Tables of pure quintic fields

TL;DR

This paper classifies and tabulates arithmetical invariants of 900 pure quintic fields and their Galois closures, extending previous cubic field theories to quintic fields using Galois cohomology and class number relations.

Contribution

It generalizes Barrucand and Cohn's theory to pure quintic fields, providing a comprehensive classification and database of their arithmetic properties based on Galois cohomology.

Findings

Classification of 13 principal factorization types

Extensive database of 900 pure quintic fields

Analysis of distribution and splitting of types

Abstract

By making use of our generalization of Barrucand and Cohn's theory of principal factorizations in pure cubic fields $\mathbb{Q}(\sqrt[3]{D})$ and their Galois closures $\mathbb{Q}(\zeta_3,\sqrt[3]{D})$ with 3 possible types to pure quintic fields $L=\mathbb{Q}(\sqrt[5]{D})$ and their pure metacyclic normal fields $N=\mathbb{Q}(\zeta_5,\sqrt[5]{D})$ with 13 possible types, we compile an extensive database with arithmetical invariants of the 900 pairwise non-isomorphic fields $N$ having normalized radicands in the range $2\le D<10^3$. Our classification is based on the Galois cohomology of the unit group $U_N$, viewed as a module over the automorphism group $\mathrm{Gal}(N/K)$ of $N$ over the cyclotomic field $K=\mathbb{Q}(\zeta_5)$, by employing theorems of Hasse and Iwasawa on the Herbrand quotient of the unit norm index $(U_K:N_{N/K}(U_N))$ by the number $\#(\mathcal{P}_{N/K} / \mathcal{P}_K)$ of primitive ambiguous principal ideals, which can be interpreted as principal factors of the different $\mathfrak{D}_{N/K}$. The precise structure of the $\mathbb{F}_5$-vector space of differential principal factors is expressed in terms of norm kernels and central orthogonal idempotents. A connection with integral representation theory is established via class number relations by Parry and Walter involving the index of subfield units $(U_N:U_0)$. The statistical distribution of the 13 principal factorization types and their refined splitting into similarity classes with representative prototypes is discussed thoroughly.

Tables

Table 1. Table 1. Multiplicity of fields of species 1 ​ a 1 a 1\mathrm{a}

$t$	$0$	$1$	$2$	$3$	$4$	$5$
$m$	$1$	$4$	$16$	$64$	$256$	$1024$

Table 2. Table 2. Multiplicity of fields of species 1 ​ b 1 b 1\mathrm{b}

$u$	$v$	$0$	$1$	$2$	$3$	$4$	$5$
$0$		$0$	$1$	$3$	$13$	$51$	$205$
$1$		$0$	$4$	$12$	$52$	$204$	$820$
$2$		$0$	$16$	$48$	$208$	$816$
$3$		$0$	$64$	$192$	$832$
$4$		$0$	$256$	$768$

Table 3. Table 3. Multiplicity of fields of species 2 2 2

$u$	$v$	$0$	$1$	$2$	$3$	$4$	$5$
$0$		$0$	$0$	$1$	$3$	$13$	$51$
$1$		$1$	$0$	$4$	$12$	$52$	$204$
$2$		$4$	$0$	$16$	$48$	$208$	$816$
$3$		$16$	$0$	$64$	$192$	$832$
$4$		$64$	$0$	$256$	$768$

Table 4. Table 4. Differential principal factorization types, T, of pure metacyclic fields N 𝑁 N

T	$U$	$\eta$	$\zeta$	$A$	$I$	$R$
${\alpha }_1$	$2$	$-$	$-$	$1$	$0$	$2$
${\alpha }_2$	$2$	$-$	$-$	$1$	$1$	$1$
${\alpha }_3$	$2$	$-$	$-$	$1$	$2$	$0$
${\beta }_1$	$2$	$-$	$-$	$2$	$0$	$1$
${\beta }_2$	$2$	$-$	$-$	$2$	$1$	$0$
$\gamma$	$2$	$-$	$-$	$3$	$0$	$0$
${\delta }_1$	$1$	$\times$	$-$	$1$	$0$	$1$
${\delta }_2$	$1$	$\times$	$-$	$1$	$1$	$0$
$\epsilon$	$1$	$\times$	$-$	$2$	$0$	$0$
${\zeta }_1$	$1$	$-$	$\times$	$1$	$0$	$1$
${\zeta }_2$	$1$	$-$	$\times$	$1$	$1$	$0$
$\eta$	$1$	$-$	$\times$	$2$	$0$	$0$
$\vartheta$	$0$	$\times$	$\times$	$1$	$0$	$0$

Table 5. Table 5. Logarithmic indices E , E + 𝐸 superscript 𝐸 E,E^{+} of subfield units for DPF types, T

T	$E$	$E^{+}$	or	$E$	$E^{+}$
${\alpha }_1$	$2$	$1$
${\alpha }_2$	$1$	$0$		$3$	$1$
${\alpha }_3$	$2$	$0$
${\beta }_1$	$3$	$1$		$5$	$2$
${\beta }_2$	$4$	$1$
$\gamma$	$6$	$2$
${\delta }_1$	$2$	$1$		$4$	$2$
${\delta }_2$	$3$	$1$
$\epsilon$	$5$	$2$
${\zeta }_1$	$5$	$2$
${\zeta }_2$	$4$	$1$
$\eta$	$6$	$2$
$\vartheta$	$5$	$2$

Table 6. Table 6. 40 40 40 pure metacyclic fields with normalized radicands 2 ≤ D ≤ 52 2 𝐷 52 2\leq D\leq 52

No.	$D$	Factors	S	$f^4$	$m$	$V_L$	$V_M$	$V_N$	$E$	$(1,2,4,5)$	T	P
1	*$2$		1b	$5^2{2}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
2	$3$		1b	$5^2{3}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
3	*$5$		1a	$5^6$	$1$	$0$	$0$	$0$	$5$	$(-,-,-,\otimes )$	$\vartheta$
4	*$6$	$2\cdot 3$	1b	$5^2{2}^4{3}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
5	*$7$		2	$7^4$	$1$	$0$	$0$	$0$	$5$	$(-,-,-,\otimes )$	$\vartheta$
6	*$10$	$2\cdot 5$	1a	$5^6{2}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
7	*$11$		1b	$5^2{11}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$ℒ,{𝔎}_1$
8	$12$	$2^2\cdot 3$	1b	$5^2{2}^4{3}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
9	$13$		1b	$5^2{13}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
10	*$14$	$2\cdot 7$	1b	$5^2{2}^4{7}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
11	$15$	$3\cdot 5$	1a	$5^6{3}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
12	$17$		1b	$5^2{17}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
13	*$18$	$2\cdot 3^2$	2	$2^4{3}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
14	*$19$		1b	$5^2{19}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
15	$20$	$2^2\cdot 5$	1a	$5^6{2}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
16	$21$	$3\cdot 7$	1b	$5^2{3}^4{7}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
17	*$22$	$2\cdot 11$	1b	$5^2{2}^4{11}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$2\cdot 5,𝒦$
18	$23$		1b	$5^2{23}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
19	$26$	$2\cdot 13$	2	$2^4{13}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
20	$28$	$2^2\cdot 7$	1b	$5^2{2}^4{7}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
21	$29$		1b	$5^2{29}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
22	*$30$	$2\cdot 3\cdot 5$	1a	$5^6{2}^4{3}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
23	*$31$		1b	$5^2{31}^4$	$1$	$2$	$3$	$5$	$2$	$(-,-,\otimes ,-)$	${\alpha }_1$	${𝔎}_1,{𝔎}_2$
24	*$33$	$3\cdot 11$	1b	$5^2{3}^4{11}^4$	$3$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_2$
25	$34$	$2\cdot 17$	1b	$5^2{2}^4{17}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
26	*$35$	$5\cdot 7$	1a	$5^6{7}^4$	$4$	$0$	$0$	$1$	$6$	$(-,-,-,\otimes )$	$\eta$
27	$37$		1b	$5^2{37}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
28	*$38$	$2\cdot 19$	1b	$5^2{2}^4{19}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$5,𝒦$
29	$39$	$3\cdot 13$	1b	$5^2{3}^4{13}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
30	$40$	$2^3\cdot 5$	1a	$5^6{2}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
31	$41$		1b	$5^2{41}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$ℒ,{𝔎}_2$
32	*$42$	$2\cdot 3\cdot 7$	1b	$5^2{2}^4{3}^4{7}^4$	$12$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2\cdot 5,3\cdot 5^2$
33	$43$		2	${43}^4$	$1$	$0$	$0$	$0$	$5$	$(-,-,-,\otimes )$	$\vartheta$
34	$44$	$2^2\cdot 11$	1b	$5^2{2}^4{11}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$2\cdot 5,𝒦$
35	$45$	$3^2\cdot 5$	1a	$5^6{3}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
36	$46$	$2\cdot 23$	1b	$5^2{2}^4{23}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
37	$47$		1b	$5^2{47}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
38	$48$	$2^4\cdot 3$	1b	$5^2{2}^4{3}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
39	$51$	$3\cdot 17$	2	$3^4{17}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
40	$52$	$2^2\cdot 13$	1b	$5^2{2}^4{13}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$

Table 7. Table 7. 45 45 45 pure metacyclic fields with normalized radicands 53 ≤ D ≤ 104 53 𝐷 104 53\leq D\leq 104

No.	$D$	Factors	S	$f^4$	$m$	$V_L$	$V_M$	$V_N$	$E$	$(1,2,4,5)$	T	P
41	$53$		1b	$5^2{53}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
42	*$55$	$5\cdot 11$	1a	$5^6{11}^4$	$4$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
43	$56$	$2^3\cdot 7$	1b	$5^2{2}^4{7}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
44	*$57$	$3\cdot 19$	2	$3^4{19}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$𝒦$
45	$58$	$2\cdot 29$	1b	$5^2{2}^4{29}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$29\cdot 5^2,𝒦$
46	$59$		1b	$5^2{59}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
47	$60$	$2^2\cdot 3\cdot 5$	1a	$5^6{2}^4{3}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
48	$61$		1b	$5^2{61}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$ℒ,{𝔎}_2$
49	$62$	$2\cdot 31$	1b	$5^2{2}^4{31}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$5,𝒦$
50	$63$	$3^2\cdot 7$	1b	$5^2{3}^4{7}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
51	$65$	$5\cdot 13$	1a	$5^6{13}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
52	*$66$	$2\cdot 3\cdot 11$	1b	$5^2{2}^4{3}^4{11}^4$	$13$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$2\cdot 5,3\cdot 5^3$
53	$67$		1b	$5^2{67}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
54	$68$	$2^2\cdot 17$	2	$2^4{17}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
55	$69$	$3\cdot 23$	1b	$5^2{3}^4{23}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
56	*$70$	$2\cdot 5\cdot 7$	1a	$5^6{2}^4{7}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
57	$71$		1b	$5^2{71}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$ℒ,{𝔎}_1$
58	$73$		1b	$5^2{73}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
59	$74$	$2\cdot 37$	2	$2^4{37}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
60	$75$	$3\cdot 5^2$	1a	$5^6{3}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
61	$76$	$2^2\cdot 19$	2	$2^4{19}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$𝒦$
62	*$77$	$7\cdot 11$	1b	$5^2{7}^4{11}^4$	$4$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$11\cdot 5^3,𝒦$
63	*$78$	$2\cdot 3\cdot 13$	1b	$5^2{2}^4{3}^4{13}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$3,2\cdot 5^3$
64	$79$		1b	$5^2{79}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
65	$80$	$2^4\cdot 5$	1a	$5^6{2}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
66	*$82$	$2\cdot 41$	2	$2^4{41}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_2$
67	$83$		1b	$5^2{83}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
68	$84$	$2^2\cdot 3\cdot 7$	1b	$5^2{2}^4{3}^4{7}^4$	$12$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2\cdot 5,3\cdot 5$
69	$85$	$5\cdot 17$	1a	$5^6{17}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
70	$86$	$2\cdot 43$	1b	$5^2{2}^4{43}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
71	$87$	$3\cdot 29$	1b	$5^2{3}^4{29}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$29,ℒ$
72	$88$	$2^3\cdot 11$	1b	$5^2{2}^4{11}^4$	$3$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_2$
73	$89$		1b	$5^2{89}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
74	$90$	$2\cdot 3^2\cdot 5$	1a	$5^6{2}^4{3}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
75	$91$	$7\cdot 13$	1b	$5^2{7}^4{13}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
76	$92$	$2^2\cdot 23$	1b	$5^2{2}^4{23}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
77	$93$	$3\cdot 31$	2	$3^4{31}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
78	$94$	$2\cdot 47$	1b	$5^2{2}^4{47}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
79	*$95$	$5\cdot 19$	1a	$5^6{19}^4$	$4$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$𝒦$
80	$97$		1b	$5^2{97}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
81	$99$	$3^2\cdot 11$	2	$3^4{11}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
82	*$101$		2	${101}^4$	$1$	$1$	$2$	$4$	$5$	$(-,-,\otimes ,\otimes )$	${\zeta }_1$	${𝔎}_2$
83	$102$	$2\cdot 3\cdot 17$	1b	$5^2{2}^4{3}^4{17}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2,3\cdot 5$
84	$103$		1b	$5^2{103}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
85	$104$	$2^3\cdot 13$	1b	$5^2{2}^4{13}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$

Table 8. Table 8. 45 45 45 pure metacyclic fields with normalized radicands 105 ≤ D ≤ 155 105 𝐷 155 105\leq D\leq 155

No.	$D$	Factors	S	$f^4$	$m$	$V_L$	$V_M$	$V_N$	$E$	$(1,2,4,5)$	T	P
86	$105$	$3\cdot 5\cdot 7$	1a	$5^6{3}^4{7}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
87	$106$	$2\cdot 53$	1b	$5^2{2}^4{53}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
88	$107$		2	${107}^4$	$1$	$0$	$0$	$0$	$5$	$(-,-,-,\otimes )$	$\vartheta$
89	$109$		1b	$5^2{109}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
90	*$110$	$2\cdot 5\cdot 11$	1a	$5^6{2}^4{11}^4$	$16$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$11,ℒ$
91	$111$	$3\cdot 37$	1b	$5^2{3}^4{37}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
92	$112$	$2^4\cdot 7$	1b	$5^2{2}^4{7}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
93	$113$		1b	$5^2{113}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
94	*$114$	$2\cdot 3\cdot 19$	1b	$5^2{2}^4{3}^4{19}^4$	$13$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$2\cdot 5^3,3\cdot 5^3$
95	$115$	$5\cdot 23$	1a	$5^6{23}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
96	$116$	$2^2\cdot 29$	1b	$5^2{2}^4{29}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$29\cdot 5,𝒦$
97	$117$	$3^2\cdot 13$	1b	$5^2{3}^4{13}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
98	$118$	$2\cdot 59$	2	$2^4{59}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$𝒦$
99	$119$	$7\cdot 17$	1b	$5^2{7}^4{17}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
100	$120$	$2^3\cdot 3\cdot 5$	1a	$5^6{2}^4{3}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
101	$122$	$2\cdot 61$	1b	$5^2{2}^4{61}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$61\cdot 5^3,𝒦$
102	*$123$	$3\cdot 41$	1b	$5^2{3}^4{41}^4$	$3$	$2$	$3$	$6$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_2$
103	$124$	$2^2\cdot 31$	2	$2^4{31}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
104	*$126$	$2\cdot 3^2\cdot 7$	2	$2^4{3}^4{7}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
105	$127$		1b	$5^2{127}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
106	$129$	$3\cdot 43$	1b	$5^2{3}^4{43}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
107	$130$	$2\cdot 5\cdot 13$	1a	$5^6{2}^4{13}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
108	*$131$		1b	$5^2{131}^4$	$1$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$ℒ,{𝔎}_2$
109	*$132$	$2^2\cdot 3\cdot 11$	2	$2^4{3}^4{11}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$11,ℒ$
110	*$133$	$7\cdot 19$	1b	$5^2{7}^4{19}^4$	$4$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$7,ℒ$
111	$134$	$2\cdot 67$	1b	$5^2{2}^4{67}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
112	$136$	$2^3\cdot 17$	1b	$5^2{2}^4{17}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
113	$137$		1b	$5^2{137}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
114	$138$	$2\cdot 3\cdot 23$	1b	$5^2{2}^4{3}^4{23}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$3,2\cdot 5^2$
115	*$139$		1b	$5^2{139}^4$	$1$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$5,ℒ$
116	*$140$	$2^2\cdot 5\cdot 7$	1a	$5^6{2}^4{7}^4$	$16$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$7$
117	*$141$	$3\cdot 47$	1b	$5^2{3}^4{47}^4$	$3$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$47\cdot 5$
118	$142$	$2\cdot 71$	1b	$5^2{2}^4{71}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$71\cdot 5^2,𝒦$
119	$143$	$11\cdot 13$	2	${11}^4{13}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
120	$145$	$5\cdot 29$	1a	$5^6{29}^4$	$4$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$𝒦$
121	$146$	$2\cdot 73$	1b	$5^2{2}^4{73}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
122	$147$	$3\cdot 7^2$	1b	$5^2{3}^4{7}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
123	$148$	$2^2\cdot 37$	1b	$5^2{2}^4{37}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
124	*$149$		2	${149}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,\times )$	${\delta }_2$	$ℒ$
125	$150$	$2\cdot 3\cdot 5^2$	1a	$5^6{2}^4{3}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
126	*$151$		2	${151}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,\times )$	${\alpha }_2$	$ℒ,{𝔎}_1$
127	$152$	$2^3\cdot 19$	1b	$5^2{2}^4{19}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$19\cdot 5^2,𝒦$
128	$153$	$3^2\cdot 17$	1b	$5^2{3}^4{17}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
129	*$154$	$2\cdot 7\cdot 11$	1b	$5^2{2}^4{7}^4{11}^4$	$12$	$2$	$3$	$7$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$2\cdot {11}^2,𝒦$
130	*$155$	$5\cdot 31$	1a	$5^6{31}^4$	$4$	$2$	$3$	$5$	$2$	$(-,-,\otimes ,-)$	${\alpha }_1$	${𝔎}_1,{𝔎}_2$

Table 9. Table 9. 45 45 45 pure metacyclic fields with normalized radicands 156 ≤ D ≤ 207 156 𝐷 207 156\leq D\leq 207

No.	$D$	Factors	S	$f^4$	$m$	$V_L$	$V_M$	$V_N$	$E$	$(1,2,4,5)$	T	P
131	$156$	$2^2\cdot 3\cdot 13$	1b	$5^2{2}^4{3}^4{13}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$3,13\cdot 5^2$
132	$157$		2	${157}^4$	$1$	$0$	$0$	$0$	$5$	$(-,-,-,\otimes )$	$\vartheta$
133	$158$	$2\cdot 79$	1b	$5^2{2}^4{79}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$5,𝒦$
134	$159$	$3\cdot 53$	1b	$5^2{3}^4{53}^4$	$3$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$5$
135	$161$	$7\cdot 23$	1b	$5^2{7}^4{23}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
136	$163$		1b	$5^2{163}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
137	$164$	$2^2\cdot 41$	1b	$5^2{2}^4{41}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$41\cdot 5,𝒦$
138	$165$	$3\cdot 5\cdot 11$	1a	$5^6{3}^4{11}^4$	$16$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$5\cdot {11}^2,𝒦$
139	$166$	$2\cdot 83$	1b	$5^2{2}^4{83}^4$	$3$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$5$
140	$167$		1b	$5^2{167}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
141	$168$	$2^3\cdot 3\cdot 7$	2	$2^4{3}^4{7}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
142	$170$	$2\cdot 5\cdot 17$	1a	$5^6{2}^4{17}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
143	*$171$	$3^2\cdot 19$	1b	$5^2{3}^4{19}^4$	$3$	$1$	$2$	$4$	$5$	$(-,\times ,-,-)$	$\epsilon$	$19\cdot 5^2$
144	$172$	$2^2\cdot 43$	1b	$5^2{2}^4{43}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
145	$173$		1b	$5^2{173}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
146	*$174$	$2\cdot 3\cdot 29$	2	$2^4{3}^4{29}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$3^2\cdot 29,𝒦$
147	$175$	$5^2\cdot 7$	1a	$5^6{7}^4$	$4$	$0$	$0$	$1$	$6$	$(-,-,-,\otimes )$	$\eta$
148	$176$	$2^4\cdot 11$	2	$2^4{11}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_2$
149	$177$	$3\cdot 59$	1b	$5^2{3}^4{59}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$3,ℒ$
150	$178$	$2\cdot 89$	1b	$5^2{2}^4{89}^4$	$3$	$1$	$2$	$4$	$5$	$(-,\times ,-,-)$	$\epsilon$	$89\cdot 5^2$
151	$179$		1b	$5^2{179}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
152	*$180$	$2^2\cdot 3^2\cdot 5$	1a	$5^6{2}^4{3}^4$	$16$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$3$
153	$181$		1b	$5^2{181}^4$	$1$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$ℒ,{𝔎}_2$
154	*$182$	$2\cdot 7\cdot 13$	2	$2^4{7}^4{13}^4$	$4$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$7$
155	$183$	$3\cdot 61$	1b	$5^2{3}^4{61}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$61\cdot 5^3,𝒦$
156	$184$	$2^3\cdot 23$	1b	$5^2{2}^4{23}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
157	$185$	$5\cdot 37$	1a	$5^6{37}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
158	*$186$	$2\cdot 3\cdot 31$	1b	$5^2{2}^4{3}^4{31}^4$	$13$	$2$	$3$	$6$	$3$	$(-,-,\otimes ,-)$	${\beta }_1$	$5,{𝔎}_2$
159	$187$	$11\cdot 17$	1b	$5^2{11}^4{17}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$11\cdot 5,𝒦$
160	$188$	$2^2\cdot 47$	1b	$5^2{2}^4{47}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
161	*$190$	$2\cdot 5\cdot 19$	1a	$5^6{2}^4{19}^4$	$16$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$5,𝒦$
162	*$191$		1b	$5^2{191}^4$	$1$	$1$	$2$	$4$	$5$	$(-,-,\otimes ,-)$	${\beta }_1$	$5,{𝔎}_1$
163	$193$		2	${193}^4$	$1$	$0$	$0$	$0$	$5$	$(-,-,-,\otimes )$	$\vartheta$
164	$194$	$2\cdot 97$	1b	$5^2{2}^4{97}^4$	$3$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$97\cdot 5^2$
165	$195$	$3\cdot 5\cdot 13$	1a	$5^6{3}^4{13}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
166	$197$		1b	$5^2{197}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
167	$198$	$2\cdot 3^2\cdot 11$	1b	$5^2{2}^4{3}^4{11}^4$	$13$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$3\cdot 5^3,11\cdot 5^3$
168	$199$		2	${199}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,\times )$	${\delta }_2$	$ℒ$
169	$201$	$3\cdot 67$	2	$3^4{67}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
170	*$202$	$2\cdot 101$	1b	$5^2{2}^4{101}^4$	$4$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$2\cdot 5,𝒦$
171	*$203$	$7\cdot 29$	1b	$5^2{7}^4{29}^4$	$4$	$1$	$2$	$4$	$5$	$(-,\times ,-,-)$	$\epsilon$	$29\cdot 5$
172	$204$	$2^2\cdot 3\cdot 17$	1b	$5^2{2}^4{3}^4{17}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$3\cdot 5^4,17\cdot 5^3$
173	$205$	$5\cdot 41$	1a	$5^6{41}^4$	$4$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_2$
174	$206$	$2\cdot 103$	1b	$5^2{2}^4{103}^4$	$3$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$2\cdot 5^2$
175	$207$	$3^2\cdot 23$	2	$3^4{23}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$

Table 10. Table 10. 45 45 45 pure metacyclic fields with normalized radicands 208 ≤ D ≤ 259 208 𝐷 259 208\leq D\leq 259

No.	$D$	Factors	S	$f^4$	$m$	$V_L$	$V_M$	$V_N$	$E$	$(1,2,4,5)$	T	P
176	$208$	$2^4\cdot 13$	1b	$5^2{2}^4{13}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
177	*$209$	$11\cdot 19$	1b	$5^2{11}^4{19}^4$	$3$	$2$	$3$	$7$	$4$	$(-,\otimes ,\times ,-)$	${\beta }_2$	$11\cdot 5^2,{𝒦}_{\left(19\right)}$
178	*$210$	$2\cdot 3\cdot 5\cdot 7$	1a	$5^6{2}^4{3}^4{7}^4$	$64$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$7,3^2\cdot 5$
179	*$211$		1b	$5^2{211}^4$	$1$	$3$	$5$	$9$	$2$	$(-,-,\otimes ,-)$	${\delta }_1$	${𝔎}_2$
180	$212$	$2^2\cdot 53$	1b	$5^2{2}^4{53}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
181	$213$	$3\cdot 71$	1b	$5^2{3}^4{71}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$3\cdot 5^3,𝒦$
182	$214$	$2\cdot 107$	1b	$5^2{2}^4{107}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
183	$215$	$5\cdot 43$	1a	$5^6{43}^4$	$4$	$0$	$0$	$1$	$6$	$(-,-,-,\otimes )$	$\eta$
184	$217$	$7\cdot 31$	1b	$5^2{7}^4{31}^4$	$4$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$5,𝒦$
185	*$218$	$2\cdot 109$	2	$2^4{109}^4$	$1$	$1$	$2$	$4$	$5$	$(-,\times ,-,-)$	$\epsilon$	$2$
186	$219$	$3\cdot 73$	1b	$5^2{3}^4{73}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
187	$220$	$2^2\cdot 5\cdot 11$	1a	$5^6{2}^4{11}^4$	$16$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$2\cdot 5,𝒦$
188	$221$	$13\cdot 17$	1b	$5^2{13}^4{17}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
189	$222$	$2\cdot 3\cdot 37$	1b	$5^2{2}^4{3}^4{37}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$37,2\cdot 5^2$
190	$223$		1b	$5^2{223}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
191	$226$	$2\cdot 113$	2	$2^4{113}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
192	$227$		1b	$5^2{227}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
193	$228$	$2^2\cdot 3\cdot 19$	1b	$5^2{2}^4{3}^4{19}^4$	$13$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$3\cdot 5,19\cdot 5$
194	$229$		1b	$5^2{229}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
195	$230$	$2\cdot 5\cdot 23$	1a	$5^6{2}^4{23}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
196	*$231$	$3\cdot 7\cdot 11$	1b	$5^2{3}^4{7}^4{11}^4$	$12$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$11,7\cdot 5^2$
197	$232$	$2^3\cdot 29$	2	$2^4{29}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$𝒦$
198	$233$		1b	$5^2{233}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
199	$234$	$2\cdot 3^2\cdot 13$	1b	$5^2{2}^4{3}^4{13}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$3,2\cdot 5$
200	$235$	$5\cdot 47$	1a	$5^6{47}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
201	$236$	$2^2\cdot 59$	1b	$5^2{2}^4{59}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$2\cdot 5^2,𝒦$
202	$237$	$3\cdot 79$	1b	$5^2{3}^4{79}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$3\cdot 5,𝒦$
203	$238$	$2\cdot 7\cdot 17$	1b	$5^2{2}^4{7}^4{17}^4$	$12$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$5,2^4\cdot 7$
204	$239$		1b	$5^2{239}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
205	$240$	$2^4\cdot 3\cdot 5$	1a	$5^6{2}^4{3}^4$	$16$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$3$
206	$241$		1b	$5^2{241}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$ℒ,{𝔎}_2$
207	$244$	$2^2\cdot 61$	1b	$5^2{2}^4{61}^4$	$3$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
208	$245$	$5\cdot 7^2$	1a	$5^6{7}^4$	$4$	$0$	$0$	$1$	$6$	$(-,-,-,\otimes )$	$\eta$
209	$246$	$2\cdot 3\cdot 41$	1b	$5^2{2}^4{3}^4{41}^4$	$13$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$3,2\cdot 5^2$
210	*$247$	$13\cdot 19$	1b	$5^2{13}^4{19}^4$	$3$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$
211	$248$	$2^3\cdot 31$	1b	$5^2{2}^4{31}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$5,𝒦$
212	$249$	$3\cdot 83$	2	$3^4{83}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
213	$251$		2	${251}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,\times )$	${\alpha }_2$	$ℒ,{𝔎}_2$
214	$252$	$2^2\cdot 3^2\cdot 7$	1b	$5^2{2}^4{3}^4{7}^4$	$12$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$3,2\cdot 5$
215	*$253$	$11\cdot 23$	1b	$5^2{11}^4{23}^4$	$3$	$1$	$2$	$4$	$5$	$(-,-,\otimes ,-)$	${\beta }_1$	$11,{𝔎}_1$
216	$254$	$2\cdot 127$	1b	$5^2{2}^4{127}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
217	$255$	$3\cdot 5\cdot 17$	1a	$5^6{3}^4{17}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
218	$257$		2	${257}^4$	$1$	$0$	$0$	$0$	$5$	$(-,-,-,\otimes )$	$\vartheta$
219	$258$	$2\cdot 3\cdot 43$	1b	$5^2{2}^4{3}^4{43}^4$	$12$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2,3\cdot 5$
220	*$259$	$7\cdot 37$	1b	$5^2{7}^4{37}^4$	$4$	$1$	$2$	$5\ast$	$6$	$(\times ,-,-,-)$	$\gamma$

Table 11. Table 11. 45 45 45 pure metacyclic fields with normalized radicands 260 ≤ D ≤ 307 260 𝐷 307 260\leq D\leq 307

No.	$D$	Factors	S	$f^4$	$m$	$V_L$	$V_M$	$V_N$	$E$	$(1,2,4,5)$	T	P
221	$260$	$2^2\cdot 5\cdot 13$	1a	$5^6{2}^4{13}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
222	$261$	$3^2\cdot 29$	1b	$5^2{3}^4{29}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$29,ℒ$
223	$262$	$2\cdot 131$	1b	$5^2{2}^4{131}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$131\cdot 5,𝒦$
224	$263$		1b	$5^2{263}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
225	$264$	$2^3\cdot 3\cdot 11$	1b	$5^2{2}^4{3}^4{11}^4$	$13$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$2\cdot 5,2\cdot 11$
226	$265$	$5\cdot 53$	1a	$5^6{53}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
227	*$266$	$2\cdot 7\cdot 19$	1b	$5^2{2}^4{7}^4{19}^4$	$12$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$5,2^3\cdot 7$
228	$267$	$3\cdot 89$	1b	$5^2{3}^4{89}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$89\cdot 5^2,𝒦$
229	$268$	$2^2\cdot 67$	2	$2^4{67}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
230	$269$		1b	$5^2{269}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
231	$270$	$2\cdot 3^3\cdot 5$	1a	$5^6{2}^4{3}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
232	$271$		1b	$5^2{271}^4$	$1$	$1$	$2$	$4$	$5$	$(-,-,\otimes ,-)$	${\beta }_1$	$5,{𝔎}_1$
233	$272$	$2^4\cdot 17$	1b	$5^2{2}^4{17}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
234	*$273$	$3\cdot 7\cdot 13$	1b	$5^2{3}^4{7}^4{13}^4$	$12$	$2$	$4$	$8$	$5$	$(\times ,-,-,-)$	$\epsilon$	$7\cdot {13}^2\cdot 5^4$
235	$274$	$2\cdot 137$	2	$2^4{137}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
236	*$275$	$5^2\cdot 11$	1a	$5^6{11}^4$	$4$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_2$
237	*$276$	$2^2\cdot 3\cdot 23$	2	$2^4{3}^4{23}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
238	$277$		1b	$5^2{277}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
239	$278$	$2\cdot 139$	1b	$5^2{2}^4{139}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$139\cdot 5,𝒦$
240	$279$	$3^2\cdot 31$	1b	$5^2{3}^4{31}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$5,𝒦$
241	$280$	$2^3\cdot 5\cdot 7$	1a	$5^6{2}^4{7}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
242	*$281$		1b	$5^2{281}^4$	$1$	$3$	$5\ast$	$9\ast$	$2$	$(-,-,\otimes ,-)$	${\alpha }_1$	${𝔎}_1,{𝔎}_2$
243	$282$	$2\cdot 3\cdot 47$	2	$2^4{3}^4{47}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
244	$283$		1b	$5^2{283}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
245	$284$	$2^2\cdot 71$	1b	$5^2{2}^4{71}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$2\cdot 5^3,𝒦$
246	*$285$	$3\cdot 5\cdot 19$	1a	$5^6{3}^4{19}^4$	$16$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$
247	*$286$	$2\cdot 11\cdot 13$	1b	$5^2{2}^4{11}^4{13}^4$	$13$	$2$	$3$	$7$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	${11}^2\cdot 5,𝒦$
248	*$287$	$7\cdot 41$	1b	$5^2{7}^4{41}^4$	$4$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
249	*$290$	$2\cdot 5\cdot 29$	1a	$5^6{2}^4{29}^4$	$16$	$1$	$2$	$4$	$5$	$(-,\times ,-,-)$	$\epsilon$	$29$
250	$291$	$3\cdot 97$	1b	$5^2{3}^4{97}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
251	$292$	$2^2\cdot 73$	1b	$5^2{2}^4{73}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
252	$293$		2	${293}^4$	$1$	$0$	$0$	$0$	$5$	$(-,-,-,\otimes )$	$\vartheta$
253	$294$	$2\cdot 3\cdot 7^2$	1b	$5^2{2}^4{3}^4{7}^4$	$12$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$3,7^2\cdot 5$
254	$295$	$5\cdot 59$	1a	$5^6{59}^4$	$4$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$𝒦$
255	$296$	$2^3\cdot 37$	1b	$5^2{2}^4{37}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
256	$297$	$3^3\cdot 11$	1b	$5^2{3}^4{11}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$11\cdot 5,𝒦$
257	*$298$	$2\cdot 149$	1b	$5^2{2}^4{149}^4$	$4$	$1$	$2$	$4$	$5$	$(-,\times ,-,-)$	$\epsilon$	$5$
258	$299$	$13\cdot 23$	2	${13}^4{23}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
259	*$301$	$7\cdot 43$	2	$7^4{43}^4$	$4$	$0$	$0$	$1$	$6$	$(-,-,-,\otimes )$	$\eta$
260	*$302$	$2\cdot 151$	1b	$5^2{2}^4{151}^4$	$4$	$1$	$2$	$4$	$5$	$(-,-,\otimes ,-)$	${\beta }_1$	$2,{𝔎}_1$
261	$303$	$3\cdot 101$	1b	$5^2{3}^4{101}^4$	$4$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$101\cdot 5,𝒦$
262	$304$	$2^4\cdot 19$	1b	$5^2{2}^4{19}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$19\cdot 5^4,𝒦$
263	$305$	$5\cdot 61$	1a	$5^6{61}^4$	$4$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
264	$306$	$2\cdot 3^2\cdot 17$	1b	$5^2{2}^4{3}^4{17}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$3\cdot 5,3\cdot 17$
265	$307$		2	${307}^4$	$1$	$0$	$0$	$0$	$5$	$(-,-,-,\otimes )$	$\vartheta$

Table 12. Table 12. 45 45 45 pure metacyclic fields with normalized radicands 308 ≤ D ≤ 357 308 𝐷 357 308\leq D\leq 357

No.	$D$	Factors	S	$f^4$	$m$	$V_L$	$V_M$	$V_N$	$E$	$(1,2,4,5)$	T	P
266	$308$	$2^2\cdot 7\cdot 11$	1b	$5^2{2}^4{7}^4{11}^4$	$12$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$11\cdot 5,7\cdot 5^2$
267	$309$	$3\cdot 103$	1b	$5^2{3}^4{103}^4$	$3$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$103\cdot 5^2$
268	$310$	$2\cdot 5\cdot 31$	1a	$5^6{2}^4{31}^4$	$16$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$5,𝒦$
269	$311$		1b	$5^2{311}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$ℒ,{𝔎}_2$
270	$312$	$2^3\cdot 3\cdot 13$	1b	$5^2{2}^4{3}^4{13}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$5,2^4\cdot 13$
271	$313$		1b	$5^2{313}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
272	$314$	$2\cdot 157$	1b	$5^2{2}^4{157}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
273	$315$	$3^2\cdot 5\cdot 7$	1a	$5^6{3}^4{7}^4$	$16$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$7$
274	$316$	$2^2\cdot 79$	1b	$5^2{2}^4{79}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$2\cdot 5,𝒦$
275	$317$		1b	$5^2{317}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
276	$318$	$2\cdot 3\cdot 53$	2	$2^4{3}^4{53}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
277	*$319$	$11\cdot 29$	1b	$5^2{11}^4{29}^4$	$3$	$2$	$2$	$5$	$2$	$(-,\otimes ,(\times ),-)$	${\alpha }_3$	${𝒦}_{\left(11\right)},{𝒦}_{\left(29\right)}$
278	$321$	$3\cdot 107$	1b	$5^2{3}^4{107}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
279	$322$	$2\cdot 7\cdot 23$	1b	$5^2{2}^4{7}^4{23}^4$	$12$	$2$	$4$	$8$	$5$	$(\times ,-,-,-)$	$\epsilon$	$7\cdot {23}^2\cdot 5^3$
280	$323$	$17\cdot 19$	1b	$5^2{17}^4{19}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$19\cdot 5^2,𝒦$
281	$325$	$5^2\cdot 13$	1a	$5^6{13}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
282	$326$	$2\cdot 163$	2	$2^4{163}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
283	$327$	$3\cdot 109$	1b	$5^2{3}^4{109}^4$	$3$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$
284	$328$	$2^3\cdot 41$	1b	$5^2{2}^4{41}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$2\cdot 41\cdot 5,𝒦$
285	*$329$	$7\cdot 47$	1b	$5^2{7}^4{47}^4$	$4$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$47$
286	*$330$	$2\cdot 3\cdot 5\cdot 11$	1a	$5^6{2}^4{3}^4{11}^4$	$64$	$2$	$4$	$9$	$6$	$(-,-,\times ,-)$	$\gamma$	$3\cdot 11,5\cdot {11}^3$
287	$331$		1b	$5^2{331}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$ℒ,{𝔎}_1$
288	$332$	$2^2\cdot 83$	2	$2^4{83}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
289	$333$	$3^2\cdot 37$	1b	$5^2{3}^4{37}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
290	$334$	$2\cdot 167$	1b	$5^2{2}^4{167}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
291	$335$	$5\cdot 67$	1a	$5^6{67}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
292	$336$	$2^4\cdot 3\cdot 7$	1b	$5^2{2}^4{3}^4{7}^4$	$12$	$2$	$4$	$8$	$5$	$(\times ,-,-,-)$	$\epsilon$	$2\cdot 5^4$
293	$337$		1b	$5^2{337}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
294	$339$	$3\cdot 113$	1b	$5^2{3}^4{113}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
295	$340$	$2^2\cdot 5\cdot 17$	1a	$5^6{2}^4{17}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
296	*$341$	$11\cdot 31$	1b	$5^2{11}^4{31}^4$	$3$	$3$	$5$	$9$	$2$	$(-,-,\otimes ,-)$	${\alpha }_1$	${𝔎}_{\left(11\right),1}{𝔎}_{\left(31\right),2}^3,$
												${𝔎}_{\left(11\right),2}{𝔎}_{\left(31\right),1}$
297	$342$	$2\cdot 3^2\cdot 19$	1b	$5^2{2}^4{3}^4{19}^4$	$13$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$19,2\cdot 5^2$
298	$344$	$2^3\cdot 43$	1b	$5^2{2}^4{43}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
299	$345$	$3\cdot 5\cdot 23$	1a	$5^6{3}^4{23}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
300	$346$	$2\cdot 173$	1b	$5^2{2}^4{173}^4$	$3$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$173\cdot 5$
301	$347$		1b	$5^2{347}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
302	*$348$	$2^2\cdot 3\cdot 29$	1b	$5^2{2}^4{3}^4{29}^4$	$13$	$2$	$4$	$8$	$5$	$(-,\times ,-,-)$	$\epsilon$	$2\cdot 3\cdot 5$
303	$349$		2	${349}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,\times )$	${\delta }_2$	$ℒ$
304	$350$	$2\cdot 5^2\cdot 7$	1a	$5^6{2}^4{7}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
305	$351$	$3^3\cdot 13$	2	$3^4{13}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
306	$353$		1b	$5^2{353}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
307	$354$	$2\cdot 3\cdot 59$	1b	$5^2{2}^4{3}^4{59}^4$	$13$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$5,2\cdot 3^2$
308	$355$	$5\cdot 71$	1a	$5^6{71}^4$	$4$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
309	$356$	$2^2\cdot 89$	1b	$5^2{2}^4{89}^4$	$3$	$1$	$2$	$4$	$5$	$(-,\times ,-,-)$	$\epsilon$	$89\cdot 5^4$
310	$357$	$3\cdot 7\cdot 17$	2	$3^4{7}^4{17}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$

Table 13. Table 13. 45 45 45 pure metacyclic fields with normalized radicands 358 ≤ D ≤ 408 358 𝐷 408 358\leq D\leq 408

No.	$D$	Factors	S	$f^4$	$m$	$V_L$	$V_M$	$V_N$	$E$	$(1,2,4,5)$	T	P
311	$358$	$2\cdot 179$	1b	$5^2{2}^4{179}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$2,ℒ$
312	$359$		1b	$5^2{359}^4$	$1$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$5,ℒ$
313	$360$	$2^3\cdot 3^2\cdot 5$	1a	$5^6{2}^4{3}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
314	$362$	$2\cdot 181$	1b	$5^2{2}^4{181}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$181\cdot 5,𝒦$
315	$364$	$2^2\cdot 7\cdot 13$	1b	$5^2{2}^4{7}^4{13}^4$	$12$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2,7\cdot 13$
316	$365$	$5\cdot 73$	1a	$5^6{73}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
317	$366$	$2\cdot 3\cdot 61$	1b	$5^2{2}^4{3}^4{61}^4$	$13$	$2$	$3$	$6$	$3$	$(-,-,\otimes ,-)$	${\beta }_1$	$3\cdot {61}^2\cdot 5^4,{𝔎}_1$
318	$367$		1b	$5^2{367}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
319	$368$	$2^4\cdot 23$	2	$2^4{23}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
320	$369$	$3^2\cdot 41$	1b	$5^2{3}^4{41}^4$	$3$	$2$	$3$	$6$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_2$
321	$370$	$2\cdot 5\cdot 37$	1a	$5^6{2}^4{37}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
322	$371$	$7\cdot 53$	1b	$5^2{7}^4{53}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
323	$372$	$2^2\cdot 3\cdot 31$	1b	$5^2{2}^4{3}^4{31}^4$	$13$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$5,2\cdot 31$
324	$373$		1b	$5^2{373}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
325	$374$	$2\cdot 11\cdot 17$	2	$2^4{11}^4{17}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$11,ℒ$
326	$376$	$2^3\cdot 47$	2	$2^4{47}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
327	*$377$	$13\cdot 29$	1b	$5^2{13}^4{29}^4$	$3$	$2$	$3$	$6$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$𝒦$
328	$378$	$2\cdot 3^3\cdot 7$	1b	$5^2{2}^4{3}^4{7}^4$	$12$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2\cdot 5^2,3^3\cdot 7^2$
329	*$379$		1b	$5^2{379}^4$	$1$	$1$	$2$	$4$	$5$	$(-,\times ,-,-)$	$\epsilon$	$5$
330	$380$	$2^2\cdot 5\cdot 19$	1a	$5^6{2}^4{19}^4$	$16$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$19,ℒ$
331	$381$	$3\cdot 127$	1b	$5^2{3}^4{127}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
332	$382$	$2\cdot 191$	2	$2^4{191}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
333	$383$		1b	$5^2{383}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
334	*$385$	$5\cdot 7\cdot 11$	1a	$5^6{7}^4{11}^4$	$16$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$7\cdot {11}^2,𝒦$
335	$386$	$2\cdot 193$	1b	$5^2{2}^4{193}^4$	$4$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$2$
336	$387$	$3^2\cdot 43$	1b	$5^2{3}^4{43}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
337	$388$	$2^2\cdot 97$	1b	$5^2{2}^4{97}^4$	$3$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$97\cdot 5^4$
338	$389$		1b	$5^2{389}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
339	*$390$	$2\cdot 3\cdot 5\cdot 13$	1a	$5^6{2}^4{3}^4{13}^4$	$64$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2,5$
340	$391$	$17\cdot 23$	1b	$5^2{17}^4{23}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
341	$393$	$3\cdot 131$	2	$3^4{131}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
342	$394$	$2\cdot 197$	1b	$5^2{2}^4{197}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
343	$395$	$5\cdot 79$	1a	$5^6{79}^4$	$4$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$𝒦$
344	$396$	$2^2\cdot 3^2\cdot 11$	1b	$5^2{2}^4{3}^4{11}^4$	$13$	$2$	$3$	$6$	$3$	$(-,-,\otimes ,-)$	${\beta }_1$	$11\cdot 5^2,{𝔎}_2$
345	$397$		1b	$5^2{397}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
346	*$398$	$2\cdot 199$	1b	$5^2{2}^4{199}^4$	$4$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$5,𝒦$
347	*$399$	$3\cdot 7\cdot 19$	2	$3^4{7}^4{19}^4$	$4$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$3\cdot {19}^2,𝒦$
348	*$401$		2	${401}^4$	$1$	$2$	$3$	$5$	$2$	$(-,-,\otimes ,\times )$	${\alpha }_1$	${𝔎}_1,{𝔎}_2$
349	$402$	$2\cdot 3\cdot 67$	1b	$5^2{2}^4{3}^4{67}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$3\cdot 5^3,2\cdot 5^3$
350	$403$	$13\cdot 31$	1b	$5^2{13}^4{31}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$5,𝒦$
351	$404$	$2^2\cdot 101$	1b	$5^2{2}^4{101}^4$	$4$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$2\cdot 5,𝒦$
352	$405$	$3^4\cdot 5$	1a	$5^6{3}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
353	$406$	$2\cdot 7\cdot 29$	1b	$5^2{2}^4{7}^4{29}^4$	$12$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$7,2\cdot 5^3$
354	$407$	$11\cdot 37$	2	${11}^4{37}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_2$
355	$408$	$2^3\cdot 3\cdot 17$	1b	$5^2{2}^4{3}^4{17}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$17,3\cdot 5^3$

Table 14. Table 14. 45 45 45 pure metacyclic fields with normalized radicands 409 ≤ D ≤ 458 409 𝐷 458 409\leq D\leq 458

No.	$D$	Factors	S	$f^4$	$m$	$V_L$	$V_M$	$V_N$	$E$	$(1,2,4,5)$	T	P
356	$409$		1b	$5^2{409}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
357	$410$	$2\cdot 5\cdot 41$	1a	$5^6{2}^4{41}^4$	$16$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$2\cdot {41}^2,𝒦$
358	$411$	$3\cdot 137$	1b	$5^2{3}^4{137}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
359	$412$	$2^2\cdot 103$	1b	$5^2{2}^4{103}^4$	$3$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$2\cdot 5^2$
360	$413$	$7\cdot 59$	1b	$5^2{7}^4{59}^4$	$4$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$59\cdot 5,𝒦$
361	$414$	$2\cdot 3^2\cdot 23$	1b	$5^2{2}^4{3}^4{23}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2,3$
362	$415$	$5\cdot 83$	1a	$5^6{83}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
363	$417$	$3\cdot 139$	1b	$5^2{3}^4{139}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$139\cdot 5,𝒦$
364	*$418$	$2\cdot 11\cdot 19$	2	$2^4{11}^4{19}^4$	$3$	$2$	$3$	$6$	$3$	$(-,\otimes ,\otimes ,-)$	${\alpha }_2$	${𝒦}_{\left(19\right)},{𝔎}_{\left(11\right),1}$
365	$419$		1b	$5^2{419}^4$	$1$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$5,ℒ$
366	$420$	$2^2\cdot 3\cdot 5\cdot 7$	1a	$5^6{2}^4{3}^4{7}^4$	$64$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2,3^2\cdot 7$
367	*$421$		1b	$5^2{421}^4$	$1$	$1$	$2$	$3$	$4$	$(-,-,\otimes ,-)$	${\delta }_1$	${𝔎}_2$
368	*$422$	$2\cdot 211$	1b	$5^2{2}^4{211}^4$	$3$	$1$	$2$	$4$	$5$	$(-,-,\times ,-)$	$\epsilon$	$2\cdot 5^2$
369	$423$	$3^2\cdot 47$	1b	$5^2{3}^4{47}^4$	$3$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$3\cdot 5^3$
370	$424$	$2^3\cdot 53$	2	$2^4{53}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
371	$425$	$5^2\cdot 17$	1a	$5^6{17}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
372	$426$	$2\cdot 3\cdot 71$	2	$2^4{3}^4{71}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$2^2\cdot 71,𝒦$
373	$427$	$7\cdot 61$	1b	$5^2{7}^4{61}^4$	$4$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$61\cdot 5^2,𝒦$
374	$428$	$2^2\cdot 107$	1b	$5^2{2}^4{107}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
375	$429$	$3\cdot 11\cdot 13$	1b	$5^2{3}^4{11}^4{13}^4$	$13$	$2$	$3$	$7$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$13\cdot 5,𝒦$
376	$430$	$2\cdot 5\cdot 43$	1a	$5^6{2}^4{43}^4$	$16$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$43$
377	$431$		1b	$5^2{431}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$ℒ,{𝔎}_1$
378	$433$		1b	$5^2{433}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
379	$434$	$2\cdot 7\cdot 31$	1b	$5^2{2}^4{7}^4{31}^4$	$12$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$5,2\cdot {31}^2$
380	$435$	$3\cdot 5\cdot 29$	1a	$5^6{3}^4{29}^4$	$16$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$3\cdot 5^3,𝒦$
381	$436$	$2^2\cdot 109$	1b	$5^2{2}^4{109}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$109,ℒ$
382	$437$	$19\cdot 23$	1b	$5^2{19}^4{23}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$19\cdot 5^2,𝒦$
383	$438$	$2\cdot 3\cdot 73$	1b	$5^2{2}^4{3}^4{73}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$3,5$
384	$439$		1b	$5^2{439}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
385	$440$	$2^3\cdot 5\cdot 11$	1a	$5^6{2}^4{11}^4$	$16$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$2\cdot 5,𝒦$
386	$442$	$2\cdot 13\cdot 17$	1b	$5^2{2}^4{13}^4{17}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2\cdot 5^4,13\cdot 5^2$
387	$443$		2	${443}^4$	$1$	$0$	$0$	$0$	$5$	$(-,-,-,\otimes )$	$\vartheta$
388	$444$	$2^2\cdot 3\cdot 37$	1b	$5^2{2}^4{3}^4{37}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$5,2^3\cdot 3$
389	$445$	$5\cdot 89$	1a	$5^6{89}^4$	$4$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$𝒦$
390	$446$	$2\cdot 223$	1b	$5^2{2}^4{223}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
391	$447$	$3\cdot 149$	1b	$5^2{3}^4{149}^4$	$4$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$149\cdot 5,𝒦$
392	$449$		2	${449}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,\times )$	${\delta }_2$	$ℒ$
393	*$451$	$11\cdot 41$	2	${11}^4{41}^4$	$1$	$2$	$3$	$6$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	${𝒦}_{\left(11\right)}{𝒦}_{\left(41\right)}^4,$
												${𝔎}_{\left(11\right),1}{𝔎}_{\left(41\right),2}^3$
394	$452$	$2^2\cdot 113$	1b	$5^2{2}^4{113}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
395	$453$	$3\cdot 151$	1b	$5^2{3}^4{151}^4$	$4$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$3\cdot 5^2,𝒦$
396	$454$	$2\cdot 227$	1b	$5^2{2}^4{227}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
397	$455$	$5\cdot 7\cdot 13$	1a	$5^6{7}^4{13}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
398	$456$	$2^3\cdot 3\cdot 19$	1b	$5^2{2}^4{3}^4{19}^4$	$13$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$2\cdot 5,19$
399	$457$		2	${457}^4$	$1$	$0$	$0$	$0$	$5$	$(-,-,-,\otimes )$	$\vartheta$
400	$458$	$2\cdot 229$	1b	$5^2{2}^4{229}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$229\cdot 5,𝒦$

Table 15. Table 15. 45 45 45 pure metacyclic fields with normalized radicands 459 ≤ D ≤ 508 459 𝐷 508 459\leq D\leq 508

No.	$D$	Factors	S	$f^4$	$m$	$V_L$	$V_M$	$V_N$	$E$	$(1,2,4,5)$	T	P
401	$459$	$3^3\cdot 17$	1b	$5^2{3}^4{17}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
402	$460$	$2^2\cdot 5\cdot 23$	1a	$5^6{2}^4{23}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
403	$461$		1b	$5^2{461}^4$	$1$	$1$	$2$	$3$	$4$	$(-,-,\otimes ,-)$	${\delta }_1$	${𝔎}_1$
404	*$462$	$2\cdot 3\cdot 7\cdot 11$	1b	$5^2{2}^4{3}^4{7}^4{11}^4$	$52$	$2$	$4$	$9$	$6$	$(-,-,\times ,-)$	$\gamma$	$5^2\cdot 7,3\cdot 5\cdot {11}^2$
405	$463$		1b	$5^2{463}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
406	$464$	$2^4\cdot 29$	1b	$5^2{2}^4{29}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$5,𝒦$
407	*$465$	$3\cdot 5\cdot 31$	1a	$5^6{3}^4{31}^4$	$16$	$2$	$3$	$7\ast$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$5,𝒦$
408	$466$	$2\cdot 233$	1b	$5^2{2}^4{233}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
409	$467$		1b	$5^2{467}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
410	$468$	$2^2\cdot 3^2\cdot 13$	2	$2^4{3}^4{13}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
411	$469$	$7\cdot 67$	1b	$5^2{7}^4{67}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
412	$470$	$2\cdot 5\cdot 47$	1a	$5^6{2}^4{47}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
413	$471$	$3\cdot 157$	1b	$5^2{3}^4{157}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
414	$472$	$2^3\cdot 59$	1b	$5^2{2}^4{59}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$2\cdot 5,𝒦$
415	*$473$	$11\cdot 43$	1b	$5^2{11}^4{43}^4$	$4$	$2$	$3$	$7\ast$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$11,ℒ$
416	$474$	$2\cdot 3\cdot 79$	2	$2^4{3}^4{79}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$3,𝒦$
417	$475$	$5^2\cdot 19$	1a	$5^6{19}^4$	$4$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$𝒦$
418	$476$	$2^2\cdot 7\cdot 17$	2	$2^4{7}^4{17}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
419	$477$	$3^2\cdot 53$	1b	$5^2{3}^4{53}^4$	$3$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$3\cdot 5$
420	$478$	$2\cdot 239$	1b	$5^2{2}^4{239}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$239,ℒ$
421	$479$		1b	$5^2{479}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
422	$481$	$13\cdot 37$	1b	$5^2{13}^4{37}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
423	*$482$	$2\cdot 241$	2	$2^4{241}^4$	$1$	$1$	$2$	$4$	$5$	$(-,-,\otimes ,-)$	${\beta }_1$	$241,{𝔎}_2$
424	$483$	$3\cdot 7\cdot 23$	1b	$5^2{3}^4{7}^4{23}^4$	$12$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$7,3\cdot 23$
425	$485$	$5\cdot 97$	1a	$5^6{97}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
426	$487$		1b	$5^2{487}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
427	$488$	$2^3\cdot 61$	1b	$5^2{2}^4{61}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$2\cdot 5^2,𝒦$
428	$489$	$3\cdot 163$	1b	$5^2{3}^4{163}^4$	$3$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$3\cdot 5^2$
429	$490$	$2\cdot 5\cdot 7^2$	1a	$5^6{2}^4{7}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
430	$491$		1b	$5^2{491}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$ℒ,{𝔎}_1$
431	$492$	$2^2\cdot 3\cdot 41$	1b	$5^2{2}^4{3}^4{41}^4$	$13$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$3,2\cdot 5^2$
432	$493$	$17\cdot 29$	2	${17}^4{29}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$𝒦$
433	$494$	$2\cdot 13\cdot 19$	1b	$5^2{2}^4{13}^4{19}^4$	$13$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$2,13\cdot 5^2$
434	$495$	$3^2\cdot 5\cdot 11$	1a	$5^6{3}^4{11}^4$	$16$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$11,ℒ$
435	$496$	$2^4\cdot 31$	1b	$5^2{2}^4{31}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$5,𝒦$
436	$497$	$7\cdot 71$	1b	$5^2{7}^4{71}^4$	$4$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$7\cdot 5^3,𝒦$
437	$498$	$2\cdot 3\cdot 83$	1b	$5^2{2}^4{3}^4{83}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2^3\cdot 3,2\cdot 5^2$
438	$499$		2	${499}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,\times )$	${\delta }_2$	$ℒ$
439	$501$	$3\cdot 167$	2	$3^4{167}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
440	*$502$	$2\cdot 251$	1b	$5^2{2}^4{251}^4$	$4$	$2\ast$	$4\ast$	$8\ast$	$5$	$(-,-,\otimes ,-)$	${\beta }_1$	$251\cdot 5,{𝔎}_2$
441	$503$		1b	$5^2{503}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
442	$504$	$2^3\cdot 3^2\cdot 7$	1b	$5^2{2}^4{3}^4{7}^4$	$12$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$7,2\cdot 5^2$
443	*$505$	$5\cdot 101$	1a	$5^6{101}^4$	$4$	$1$	$1$	$3$	$4$	$(-,-,(\times ),\otimes )$	${\zeta }_2$	$𝒦$
444	$506$	$2\cdot 11\cdot 23$	1b	$5^2{2}^4{11}^4{23}^4$	$13$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$23,2\cdot 5$
445	$508$	$2^2\cdot 127$	1b	$5^2{2}^4{127}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$

Table 16. Table 16. 45 45 45 pure metacyclic fields with normalized radicands 509 ≤ D ≤ 556 509 𝐷 556 509\leq D\leq 556

No.	$D$	Factors	S	$f^4$	$m$	$V_L$	$V_M$	$V_N$	$E$	$(1,2,4,5)$	T	P
446	$509$		1b	$5^2{509}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
447	$510$	$2\cdot 3\cdot 5\cdot 17$	1a	$5^6{2}^4{3}^4{17}^4$	$64$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2\cdot 3,2\cdot 17$
448	$511$	$7\cdot 73$	1b	$5^2{7}^4{73}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
449	$513$	$3^3\cdot 19$	1b	$5^2{3}^4{19}^4$	$3$	$1$	$2$	$4$	$5$	$(-,\times ,-,-)$	$\epsilon$	$3\cdot 5^2$
450	$514$	$2\cdot 257$	1b	$5^2{2}^4{257}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
451	$515$	$5\cdot 103$	1a	$5^6{103}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
452	$516$	$2^2\cdot 3\cdot 43$	1b	$5^2{2}^4{3}^4{43}^4$	$12$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2\cdot 3^2,2\cdot 5^4$
453	*$517$	$11\cdot 47$	1b	$5^2{11}^4{47}^4$	$3$	$2$	$3$	$6$	$3$	$(-,-,\otimes ,-)$	${\beta }_1$	$11\cdot 5^2,{𝔎}_2$
454	$518$	$2\cdot 7\cdot 37$	2	$2^4{7}^4{37}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
455	$519$	$3\cdot 173$	1b	$5^2{3}^4{173}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
456	$520$	$2^3\cdot 5\cdot 13$	1a	$5^6{2}^4{13}^4$	$16$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$2$
457	$521$		1b	$5^2{521}^4$	$1$	$1$	$2$	$3$	$4$	$(-,-,\otimes ,-)$	${\delta }_1$	${𝔎}_2$
458	$522$	$2\cdot 3^2\cdot 29$	1b	$5^2{2}^4{3}^4{29}^4$	$13$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$2,29\cdot 5$
459	$523$		1b	$5^2{523}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
460	$524$	$2^2\cdot 131$	2	$2^4{131}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
461	$525$	$3\cdot 5^2\cdot 7$	1a	$5^6{3}^4{7}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
462	$526$	$2\cdot 263$	2	$2^4{263}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
463	$527$	$17\cdot 31$	1b	$5^2{17}^4{31}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$5,𝒦$
464	$528$	$2^4\cdot 3\cdot 11$	1b	$5^2{2}^4{3}^4{11}^4$	$13$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$2\cdot 5,3\cdot 5^3$
465	$530$	$2\cdot 5\cdot 53$	1a	$5^6{2}^4{53}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
466	$531$	$3^2\cdot 59$	1b	$5^2{3}^4{59}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$3,ℒ$
467	*$532$	$2^2\cdot 7\cdot 19$	2	$2^4{7}^4{19}^4$	$4$	$1$	$2$	$4$	$5$	$(-,\times ,-,-)$	$\epsilon$	$2\cdot 7$
468	$533$	$13\cdot 41$	1b	$5^2{13}^4{41}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$13\cdot 5^2,𝒦$
469	$534$	$2\cdot 3\cdot 89$	1b	$5^2{2}^4{3}^4{89}^4$	$13$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$3,89\cdot 5$
470	$535$	$5\cdot 107$	1a	$5^6{107}^4$	$4$	$0$	$0$	$1$	$6$	$(-,-,-,\otimes )$	$\eta$
471	$536$	$2^3\cdot 67$	1b	$5^2{2}^4{67}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
472	$537$	$3\cdot 179$	1b	$5^2{3}^4{179}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$3\cdot 5^2,𝒦$
473	$538$	$2\cdot 269$	1b	$5^2{2}^4{269}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$2,ℒ$
474	$539$	$7^2\cdot 11$	1b	$5^2{7}^4{11}^4$	$4$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$11\cdot 5,𝒦$
475	$540$	$2^2\cdot 3^3\cdot 5$	1a	$5^6{2}^4{3}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
476	$541$		1b	$5^2{541}^4$	$1$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
477	$542$	$2\cdot 271$	1b	$5^2{2}^4{271}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$5,𝒦$
478	$543$	$3\cdot 181$	2	$3^4{181}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
479	$545$	$5\cdot 109$	1a	$5^6{109}^4$	$4$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$𝒦$
480	*$546$	$2\cdot 3\cdot 7\cdot 13$	1b	$5^2{2}^4{3}^4{7}^4{13}^4$	$52$	$2$	$4$	$9$	$6$	$(\times ,-,-,-)$	$\gamma$	$3\cdot 5\cdot 13,5^2\cdot 7\cdot 13$
481	$547$		1b	$5^2{547}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
482	$548$	$2^2\cdot 137$	1b	$5^2{2}^4{137}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
483	$549$	$3^2\cdot 61$	2	$3^4{61}^4$	$3$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
484	*$550$	$2\cdot 5^2\cdot 11$	1a	$5^6{2}^4{11}^4$	$16$	$2$	$3$	$6$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
485	*$551$	$19\cdot 29$	2	${19}^4{29}^4$	$1$	$2$	$2$	$5$	$2$	$(-,\otimes ,-,-)$	${\alpha }_3$	${𝒦}_{\left(19\right)},{𝒦}_{\left(29\right)}$
486	$552$	$2^3\cdot 3\cdot 23$	1b	$5^2{2}^4{3}^4{23}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$23,3\cdot 5$
487	$553$	$7\cdot 79$	1b	$5^2{7}^4{79}^4$	$4$	$1$	$2$	$4$	$5$	$(-,\times ,-,-)$	$\epsilon$	$5$
488	$554$	$2\cdot 277$	1b	$5^2{2}^4{277}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
489	$555$	$3\cdot 5\cdot 37$	1a	$5^6{3}^4{37}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
490	$556$	$2^2\cdot 139$	1b	$5^2{2}^4{139}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$2\cdot 5^2,𝒦$

Table 17. Table 17. 45 45 45 pure metacyclic fields with normalized radicands 557 ≤ D ≤ 604 557 𝐷 604 557\leq D\leq 604

No.	$D$	Factors	S	$f^4$	$m$	$V_L$	$V_M$	$V_N$	$E$	$(1,2,4,5)$	T	P
491	$557$		2	${557}^4$	$1$	$0$	$0$	$0$	$5$	$(-,-,-,\otimes )$	$\vartheta$
492	$558$	$2\cdot 3^2\cdot 31$	1b	$5^2{2}^4{3}^4{31}^4$	$13$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$5,3\cdot 31$
493	$559$	$13\cdot 43$	1b	$5^2{13}^4{43}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
494	$560$	$2^4\cdot 5\cdot 7$	1a	$5^6{2}^4{7}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
495	$561$	$3\cdot 11\cdot 17$	1b	$5^2{3}^4{11}^4{17}^4$	$13$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$11\cdot 5^3,17\cdot 5^4$
496	$562$	$2\cdot 281$	1b	$5^2{2}^4{281}^4$	$3$	$1$	$2$	$4$	$5$	$(-,-,\times ,-)$	$\epsilon$	$2\cdot 5$
497	$563$		1b	$5^2{563}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
498	$564$	$2^2\cdot 3\cdot 47$	1b	$5^2{2}^4{3}^4{47}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2\cdot 47,2\cdot 5^2$
499	$565$	$5\cdot 113$	1a	$5^6{113}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
500	$566$	$2\cdot 283$	1b	$5^2{2}^4{283}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
501	$567$	$3^4\cdot 7$	1b	$5^2{3}^4{7}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
502	$568$	$2^3\cdot 71$	2	$2^4{71}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
503	$569$		1b	$5^2{569}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
504	*$570$	$2\cdot 3\cdot 5\cdot 19$	1a	$5^6{2}^4{3}^4{19}^4$	$64$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$2^4\cdot 3,2^4\cdot 5$
505	$571$		1b	$5^2{571}^4$	$1$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$ℒ,{𝔎}_1$
506	$572$	$2^2\cdot 11\cdot 13$	1b	$5^2{2}^4{11}^4{13}^4$	$13$	$2$	$3$	$6$	$3$	$(-,-,\otimes ,-)$	${\beta }_1$	$2\cdot {13}^2\cdot 5^3,{𝔎}_2$
507	$573$	$3\cdot 191$	1b	$5^2{3}^4{191}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$5,𝒦$
508	*$574$	$2\cdot 7\cdot 41$	2	$2^4{7}^4{41}^4$	$4$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$41,ℒ$
509	$575$	$5^2\cdot 23$	1a	$5^6{23}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
510	$577$		1b	$5^2{577}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
511	$579$	$3\cdot 193$	1b	$5^2{3}^4{193}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
512	$580$	$2^2\cdot 5\cdot 29$	1a	$5^6{2}^4{29}^4$	$16$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$2^4\cdot 5,𝒦$
513	$581$	$7\cdot 83$	1b	$5^2{7}^4{83}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
514	$582$	$2\cdot 3\cdot 97$	2	$2^4{3}^4{97}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
515	$583$	$11\cdot 53$	1b	$5^2{11}^4{53}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$53\cdot 5,𝒦$
516	$584$	$2^3\cdot 73$	1b	$5^2{2}^4{73}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
517	$585$	$3^2\cdot 5\cdot 13$	1a	$5^6{3}^4{13}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
518	$586$	$2\cdot 293$	1b	$5^2{2}^4{293}^4$	$4$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$2$
519	$587$		1b	$5^2{587}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
520	$589$	$19\cdot 31$	1b	$5^2{19}^4{31}^4$	$3$	$2$	$2$	$5$	$2$	$(-,\otimes ,(\times ),-)$	${\alpha }_3$	${𝒦}_{\left(19\right)},{𝒦}_{\left(31\right)}$
521	*$590$	$2\cdot 5\cdot 59$	1a	$5^6{2}^4{59}^4$	$16$	$2$	$3$	$7\ast$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$5,𝒦$
522	$591$	$3\cdot 197$	1b	$5^2{3}^4{197}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
523	$592$	$2^4\cdot 37$	1b	$5^2{2}^4{37}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
524	$593$		2	${593}^4$	$1$	$0$	$0$	$0$	$5$	$(-,-,-,\otimes )$	$\vartheta$
525	$594$	$2\cdot 3^3\cdot 11$	1b	$5^2{2}^4{3}^4{11}^4$	$13$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$11,3^2\cdot 5$
526	$595$	$5\cdot 7\cdot 17$	1a	$5^6{7}^4{17}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
527	$596$	$2^2\cdot 149$	1b	$5^2{2}^4{149}^4$	$4$	$1$	$2$	$4$	$5$	$(-,\times ,-,-)$	$\epsilon$	$2\cdot 5^2$
528	$597$	$3\cdot 199$	1b	$5^2{3}^4{199}^4$	$4$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$3^4\cdot 5,𝒦$
529	$598$	$2\cdot 13\cdot 23$	1b	$5^2{2}^4{13}^4{23}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2^4\cdot 5,13\cdot 5^3$
530	$599$		2	${599}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,\times )$	${\delta }_2$	$ℒ$
531	$600$	$2^3\cdot 3\cdot 5^2$	1a	$5^6{2}^4{3}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
532	$601$		2	${601}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,\times )$	${\alpha }_2$	$ℒ,{𝔎}_2$
533	*$602$	$2\cdot 7\cdot 43$	1b	$5^2{2}^4{7}^4{43}^4$	$16$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2,7$
534	$603$	$3^2\cdot 67$	1b	$5^2{3}^4{67}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
535	$604$	$2^2\cdot 151$	1b	$5^2{2}^4{151}^4$	$4$	$1$	$2$	$4$	$5$	$(-,-,\otimes ,-)$	${\beta }_1$	$2,{𝔎}_1$

Table 18. Table 18. 45 45 45 pure metacyclic fields with normalized radicands 605 ≤ D ≤ 653 605 𝐷 653 605\leq D\leq 653

No.	$D$	Factors	S	$f^4$	$m$	$V_L$	$V_M$	$V_N$	$E$	$(1,2,4,5)$	T	P
536	$605$	$5\cdot {11}^2$	1a	$5^6{11}^4$	$4$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
537	*$606$	$2\cdot 3\cdot 101$	1b	$5^2{2}^4{3}^4{101}^4$	$12$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$2\cdot 5,101$
538	$607$		2	${607}^4$	$1$	$0$	$0$	$0$	$5$	$(-,-,-,\otimes )$	$\vartheta$
539	*$609$	$3\cdot 7\cdot 29$	1b	$5^2{3}^4{7}^4{29}^4$	$12$	$2$	$3$	$7$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$7^2\cdot 5,𝒦$
540	$610$	$2\cdot 5\cdot 61$	1a	$5^6{2}^4{61}^4$	$16$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$2\cdot 5^2,𝒦$
541	$611$	$13\cdot 47$	1b	$5^2{13}^4{47}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
542	$612$	$2^2\cdot 3^2\cdot 17$	1b	$5^2{2}^4{3}^4{17}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$17,3^2\cdot 5$
543	$613$		1b	$5^2{613}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
544	$614$	$2\cdot 307$	1b	$5^2{2}^4{307}^4$	$4$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$2$
545	$615$	$3\cdot 5\cdot 41$	1a	$5^6{3}^4{41}^4$	$16$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$3,𝒦$
546	$616$	$2^3\cdot 7\cdot 11$	1b	$5^2{2}^4{7}^4{11}^4$	$12$	$2$	$3$	$7$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$7\cdot 5^2,𝒦$
547	$617$		1b	$5^2{617}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
548	$618$	$2\cdot 3\cdot 103$	2	$2^4{3}^4{103}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
549	$619$		1b	$5^2{619}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
550	*$620$	$2^2\cdot 5\cdot 31$	1a	$5^6{2}^4{31}^4$	$16$	$2$	$4\ast$	$8\ast$	$5$	$(-,-,\otimes ,-)$	${\beta }_1$	$5,{𝔎}_1$
551	$621$	$3^3\cdot 23$	1b	$5^2{3}^4{23}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
552	$622$	$2\cdot 311$	1b	$5^2{2}^4{311}^4$	$3$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
553	$623$	$7\cdot 89$	1b	$5^2{7}^4{89}^4$	$4$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$7\cdot 5,𝒦$
554	$624$	$2^4\cdot 3\cdot 13$	2	$2^4{3}^4{13}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
555	$626$	$2\cdot 313$	2	$2^4{313}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
556	*$627$	$3\cdot 11\cdot 19$	1b	$5^2{3}^4{11}^4{19}^4$	$13$	$3$	$4$	$9$	$2$	$(-,\otimes ,(\times ),-)$	${\alpha }_3$	${𝒦}_{\left(11\right)},{𝒦}_{\left(19\right)}$
557	$628$	$2^2\cdot 157$	1b	$5^2{2}^4{157}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
558	$629$	$17\cdot 37$	1b	$5^2{17}^4{37}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
559	$630$	$2\cdot 3^2\cdot 5\cdot 7$	1a	$5^6{2}^4{3}^4{7}^4$	$64$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$3,5$
560	$631$		1b	$5^2{631}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$ℒ,{𝔎}_1$
561	$632$	$2^3\cdot 79$	2	$2^4{79}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$𝒦$
562	$633$	$3\cdot 211$	1b	$5^2{3}^4{211}^4$	$3$	$1$	$2$	$4$	$5$	$(-,-,\times ,-)$	$\epsilon$	$3\cdot 5$
563	$634$	$2\cdot 317$	1b	$5^2{2}^4{317}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
564	$635$	$5\cdot 127$	1a	$5^6{127}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
565	$636$	$2^2\cdot 3\cdot 53$	1b	$5^2{2}^4{3}^4{53}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$3^2\cdot 5,53$
566	$637$	$7^2\cdot 13$	1b	$5^2{7}^4{13}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
567	*$638$	$2\cdot 11\cdot 29$	1b	$5^2{2}^4{11}^4{29}^4$	$13$	$2$	$3$	$7$	$4$	$(-,\otimes ,(\times ),-)$	${\beta }_2$	$2^4\cdot 11\cdot 5,$
												${𝒦}_{\left(11\right)}\cdot {𝒦}_{\left(29\right)}^4$
568	$639$	$3^2\cdot 71$	1b	$5^2{3}^4{71}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$3^2\cdot 5,𝒦$
569	$641$		1b	$5^2{641}^4$	$1$	$1$	$2$	$4$	$5$	$(-,-,\otimes ,-)$	${\beta }_1$	$5,{𝔎}_2$
570	$642$	$2\cdot 3\cdot 107$	1b	$5^2{2}^4{3}^4{107}^4$	$12$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$107,3\cdot 5$
571	$643$		2	${643}^4$	$1$	$0$	$0$	$0$	$5$	$(-,-,-,\otimes )$	$\vartheta$
572	$644$	$2^2\cdot 7\cdot 23$	1b	$5^2{2}^4{7}^4{23}^4$	$12$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2\cdot 5,7\cdot 5$
573	$645$	$3\cdot 5\cdot 43$	1a	$5^6{3}^4{43}^4$	$16$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$43$
574	$646$	$2\cdot 17\cdot 19$	1b	$5^2{2}^4{17}^4{19}^4$	$13$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$2\cdot 5,17$
575	$647$		1b	$5^2{647}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
576	*$649$	$11\cdot 59$	2	${11}^4{59}^4$	$1$	$2$	$2$	$5$	$2$	$(-,\otimes ,(\times ),-)$	${\alpha }_3$	${𝒦}_{\left(11\right)},{𝒦}_{\left(59\right)}$
577	$650$	$2\cdot 5^2\cdot 13$	1a	$5^6{2}^4{13}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
578	$651$	$3\cdot 7\cdot 31$	2	$3^4{7}^4{31}^4$	$4$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$3^2\cdot 7,𝒦$
579	$652$	$2^2\cdot 163$	1b	$5^2{2}^4{163}^4$	$3$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$5$
580	$653$		1b	$5^2{653}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$

Table 19. Table 19. 44 44 44 pure metacyclic fields with normalized radicands 654 ≤ D ≤ 701 654 𝐷 701 654\leq D\leq 701

No.	$D$	Factors	S	$f^4$	$m$	$V_L$	$V_M$	$V_N$	$E$	$(1,2,4,5)$	T	P
581	$654$	$2\cdot 3\cdot 109$	1b	$5^2{2}^4{3}^4{109}^4$	$13$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$3\cdot 5,109$
582	$655$	$5\cdot 131$	1a	$5^6{131}^4$	$4$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
583	$656$	$2^4\cdot 41$	1b	$5^2{2}^4{41}^4$	$3$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
584	$657$	$3^2\cdot 73$	2	$3^4{73}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
585	$658$	$2\cdot 7\cdot 47$	1b	$5^2{2}^4{7}^4{47}^4$	$12$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2\cdot 5,47\cdot 5$
586	$659$		1b	$5^2{659}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
587	*$660$	$2^2\cdot 3\cdot 5\cdot 11$	1a	$5^6{2}^4{3}^4{11}^4$	$64$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$2\cdot 5,5\cdot 11$
588	$661$		1b	$5^2{661}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$ℒ,{𝔎}_1$
589	$662$	$2\cdot 331$	1b	$5^2{2}^4{331}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$2^4\cdot 5,𝒦$
590	$663$	$3\cdot 13\cdot 17$	1b	$5^2{3}^4{13}^4{17}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$3^2\cdot 5,{13}^2\cdot 5$
591	$664$	$2^3\cdot 83$	1b	$5^2{2}^4{83}^4$	$3$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$2^2\cdot 5$
592	*$665$	$5\cdot 7\cdot 19$	1a	$5^6{7}^4{19}^4$	$16$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$7,𝒦$
593	$666$	$2\cdot 3^2\cdot 37$	1b	$5^2{2}^4{3}^4{37}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$37,2\cdot 5^2$
594	$667$	$23\cdot 29$	1b	$5^2{23}^4{29}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$23\cdot 5^2,𝒦$
595	$668$	$2^2\cdot 167$	2	$2^4{167}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
596	$669$	$3\cdot 223$	1b	$5^2{3}^4{223}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
597	$670$	$2\cdot 5\cdot 67$	1a	$5^6{2}^4{67}^4$	$16$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$2$
598	*$671$	$11\cdot 61$	1b	$5^2{11}^4{61}^4$	$3$	$3$	$4$	$8$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	${𝒦}_{\left(11\right)}{𝒦}_{\left(61\right)},$
												${𝔎}_{\left(11\right),2}{𝔎}_{\left(61\right),1}^2$
599	$673$		1b	$5^2{673}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
600	$674$	$2\cdot 337$	2	$2^4{337}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
601	$677$		1b	$5^2{677}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
602	$678$	$2\cdot 3\cdot 113$	1b	$5^2{2}^4{3}^4{113}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2^2\cdot 5,3^4\cdot 5$
603	$679$	$7\cdot 97$	1b	$5^2{7}^4{97}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
604	$680$	$2^3\cdot 5\cdot 17$	1a	$5^6{2}^4{17}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
605	$681$	$3\cdot 227$	1b	$5^2{3}^4{227}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
606	*$682$	$2\cdot 11\cdot 31$	2	$2^4{11}^4{31}^4$	$3$	$2$	$3$	$6$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	${𝒦}_{\left(11\right)}{𝒦}_{\left(31\right)}^3,$
												${𝔎}_{\left(11\right),1}{𝔎}_{\left(31\right),2}^3$
607	$683$		1b	$5^2{683}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
608	$684$	$2^2\cdot 3^2\cdot 19$	1b	$5^2{2}^4{3}^4{19}^4$	$13$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$19\cdot 5,2\cdot 5^3$
609	$685$	$5\cdot 137$	1a	$5^6{137}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
610	$687$	$3\cdot 229$	1b	$5^2{3}^4{229}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$3\cdot 5^2,𝒦$
611	$688$	$2^4\cdot 43$	1b	$5^2{2}^4{43}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
612	$689$	$13\cdot 53$	1b	$5^2{13}^4{53}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
613	$690$	$2\cdot 3\cdot 5\cdot 23$	1a	$5^6{2}^4{3}^4{23}^4$	$64$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2,5\cdot {23}^2$
614	*$691$		1b	$5^2{691}^4$	$1$	$3$	$4$	$8$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$ℒ,{𝔎}_2$
615	$692$	$2^2\cdot 173$	1b	$5^2{2}^4{173}^4$	$3$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$2\cdot 5$
616	*$693$	$3^2\cdot 7\cdot 11$	2	$3^4{7}^4{11}^4$	$4$	$2$	$3$	$6$	$3$	$(-,-,\otimes ,-)$	${\beta }_1$	$3\cdot 7,{𝔎}_2$
617	$694$	$2\cdot 347$	1b	$5^2{2}^4{347}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
618	*$695$	$5\cdot 139$	1a	$5^6{139}^4$	$4$	$1$	$2$	$4$	$5$	$(-,\times ,-,-)$	$\epsilon$	$139$
619	$696$	$2^3\cdot 3\cdot 29$	1b	$5^2{2}^4{3}^4{29}^4$	$13$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$5,2\cdot 29$
620	$697$	$17\cdot 41$	1b	$5^2{17}^4{41}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$17\cdot 5,𝒦$
621	$698$	$2\cdot 349$	1b	$5^2{2}^4{349}^4$	$4$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$2^4\cdot 5,𝒦$
622	$699$	$3\cdot 233$	2	$3^4{233}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
623	$700$	$2^2\cdot 5^2\cdot 7$	1a	$5^6{2}^4{7}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
624	$701$		2	${701}^4$	$1$	$2$	$3$	$5$	$2$	$(-,-,\otimes ,\times )$	${\alpha }_1$	${𝔎}_1,{𝔎}_2$

Table 20. Table 20. 47 47 47 pure metacyclic fields with normalized radicands 702 ≤ D ≤ 753 702 𝐷 753 702\leq D\leq 753

No.	$D$	Factors	S	$f^4$	$m$	$V_L$	$V_M$	$V_N$	$E$	$(1,2,4,5)$	T	P
625	*$702$	$2\cdot 3^3\cdot 13$	1b	$5^2{2}^4{3}^4{13}^4$	$13$	$2$	$4$	$8$	$5$	$(\times ,-,-,-)$	$\epsilon$	$13\cdot 5^4$
626	$703$	$19\cdot 37$	1b	$5^2{19}^4{37}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$19\cdot 5^2,𝒦$
627	$705$	$3\cdot 5\cdot 47$	1a	$5^6{3}^4{47}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
628	$706$	$2\cdot 353$	1b	$5^2{2}^4{353}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
629	*$707$	$7\cdot 101$	2	$7^4{101}^4$	$4$	$1$	$1$	$3$	$4$	$(-,-,(\times ),\otimes )$	${\zeta }_2$	$𝒦$
630	$708$	$2^2\cdot 3\cdot 59$	1b	$5^2{2}^4{3}^4{59}^4$	$13$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$3\cdot 5,59\cdot 5$
631	$709$		1b	$5^2{709}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
632	*$710$	$2\cdot 5\cdot 71$	1a	$5^6{2}^4{71}^4$	$16$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_2$
633	$711$	$3^2\cdot 79$	1b	$5^2{3}^4{79}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$5,𝒦$
634	$712$	$2^3\cdot 89$	1b	$5^2{2}^4{89}^4$	$3$	$1$	$2$	$4$	$5$	$(-,\times ,-,-)$	$\epsilon$	$2^2\cdot 5$
635	$713$	$23\cdot 31$	1b	$5^2{23}^4{31}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$5,𝒦$
636	$714$	$2\cdot 3\cdot 7\cdot 17$	1b	$5^2{2}^4{3}^4{7}^4{17}^4$	$52$	$2$	$4$	$9$	$6$	$(\times ,-,-,-)$	$\gamma$	$17\cdot 5^3,2\cdot 7^2\cdot 5^3$
637	$715$	$5\cdot 11\cdot 13$	1a	$5^6{11}^4{13}^4$	$16$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$11,ℒ$
638	$716$	$2^2\cdot 179$	1b	$5^2{2}^4{179}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$179,ℒ$
639	$717$	$3\cdot 239$	1b	$5^2{3}^4{239}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$3^2\cdot 5,𝒦$
640	$718$	$2\cdot 359$	2	$2^4{359}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$𝒦$
641	$719$		1b	$5^2{719}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
642	$720$	$2^4\cdot 3^2\cdot 5$	1a	$5^6{2}^4{3}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
643	$721$	$7\cdot 103$	1b	$5^2{7}^4{103}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
644	$723$	$3\cdot 241$	1b	$5^2{3}^4{241}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$3\cdot 5,𝒦$
645	$724$	$2^2\cdot 181$	2	$2^4{181}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
646	$725$	$5^2\cdot 29$	1a	$5^6{29}^4$	$4$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$𝒦$
647	$726$	$2\cdot 3\cdot {11}^2$	2	$2^4{3}^4{11}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$2^2\cdot 3,𝒦$
648	$727$		1b	$5^2{727}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
649	$728$	$2^3\cdot 7\cdot 13$	1b	$5^2{2}^4{7}^4{13}^4$	$12$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$5,7$
650	$730$	$2\cdot 5\cdot 73$	1a	$5^6{2}^4{73}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
651	$731$	$17\cdot 43$	1b	$5^2{17}^4{43}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
652	$732$	$2^2\cdot 3\cdot 61$	2	$2^4{3}^4{61}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$2^3\cdot 3^2,𝒦$
653	$733$		1b	$5^2{733}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
654	$734$	$2\cdot 367$	1b	$5^2{2}^4{367}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
655	$735$	$3\cdot 5\cdot 7^2$	1a	$5^6{3}^4{7}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
656	$737$	$11\cdot 67$	1b	$5^2{11}^4{67}^4$	$3$	$1$	$2$	$4$	$5$	$(-,-,\otimes ,-)$	${\beta }_1$	$11,{𝔎}_1$
657	$738$	$2\cdot 3^2\cdot 41$	1b	$5^2{2}^4{3}^4{41}^4$	$13$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$3,2\cdot 5^2$
658	$739$		1b	$5^2{739}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
659	$740$	$2^2\cdot 5\cdot 37$	1a	$5^6{2}^4{37}^4$	$16$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$37$
660	$741$	$3\cdot 13\cdot 19$	1b	$5^2{3}^4{13}^4{19}^4$	$13$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$3\cdot 5^2,19\cdot 5$
661	$742$	$2\cdot 7\cdot 53$	1b	$5^2{2}^4{7}^4{53}^4$	$12$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2\cdot 5^2,7\cdot 5^2$
662	$743$		2	${743}^4$	$1$	$0$	$0$	$0$	$5$	$(-,-,-,\otimes )$	$\vartheta$
663	$744$	$2^3\cdot 3\cdot 31$	1b	$5^2{2}^4{3}^4{31}^4$	$13$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$5,3\cdot {31}^2$
664	*$745$	$5\cdot 149$	1a	$5^6{149}^4$	$4$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,\otimes )$	${\zeta }_2$	$𝒦$
665	$746$	$2\cdot 373$	1b	$5^2{2}^4{373}^4$	$3$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$2^4\cdot 5$
666	$747$	$3^2\cdot 83$	1b	$5^2{3}^4{83}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
667	$748$	$2^2\cdot 11\cdot 17$	1b	$5^2{2}^4{11}^4{17}^4$	$13$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$2\cdot 5,2\cdot 17$
668	*$749$	$7\cdot 107$	2	$7^4{107}^4$	$4$	$1$	$2$	$4$	$5$	$(-,-,-,\times )$	$\epsilon$
669	*$751$		2	${751}^4$	$1$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,\times )$	${\alpha }_2$	$ℒ,{𝔎}_1$
670	$752$	$2^4\cdot 47$	1b	$5^2{2}^4{47}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
671	$753$	$3\cdot 251$	1b	$5^2{3}^4{251}^4$	$4$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$5,𝒦$

Table 21. Table 21. 44 44 44 pure metacyclic fields with normalized radicands 754 ≤ D ≤ 799 754 𝐷 799 754\leq D\leq 799

No.	$D$	Factors	S	$f^4$	$m$	$V_L$	$V_M$	$V_N$	$E$	$(1,2,4,5)$	T	P
672	$754$	$2\cdot 13\cdot 29$	1b	$5^2{2}^4{13}^4{29}^4$	$13$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$13\cdot 5,29\cdot 5$
673	$755$	$5\cdot 151$	1a	$5^6{151}^4$	$4$	$1$	$1$	$3$	$4$	$(-,-,(\times ),\otimes )$	${\zeta }_2$	$𝒦$
674	$756$	$2^2\cdot 3^3\cdot 7$	1b	$5^2{2}^4{3}^4{7}^4$	$12$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$7,2^3\cdot 5^2$
675	$757$		2	${757}^4$	$1$	$0$	$0$	$0$	$5$	$(-,-,-,\otimes )$	$\vartheta$
676	$758$	$2\cdot 379$	1b	$5^2{2}^4{379}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$2\cdot 5^2,𝒦$
677	$759$	$3\cdot 11\cdot 23$	1b	$5^2{3}^4{11}^4{23}^4$	$13$	$2$	$3$	$6$	$3$	$(-,-,\otimes ,-)$	${\beta }_1$	$3^2\cdot 23\cdot 5,{𝔎}_2$
678	$760$	$2^3\cdot 5\cdot 19$	1a	$5^6{2}^4{19}^4$	$16$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$2^4\cdot 19,𝒦$
679	$761$		1b	$5^2{761}^4$	$1$	$2$	$3$	$5$	$2$	$(-,-,\otimes ,-)$	${\alpha }_1$	${𝔎}_1,{𝔎}_2$
680	$762$	$2\cdot 3\cdot 127$	1b	$5^2{2}^4{3}^4{127}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$127\cdot 5,3\cdot 5^3$
681	$763$	$7\cdot 109$	1b	$5^2{7}^4{109}^4$	$4$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$7\cdot 5^2,𝒦$
682	$764$	$2^2\cdot 191$	1b	$5^2{2}^4{191}^4$	$3$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_2$
683	$765$	$3^2\cdot 5\cdot 17$	1a	$5^6{3}^4{17}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
684	$766$	$2\cdot 383$	1b	$5^2{2}^4{383}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
685	$767$	$13\cdot 59$	1b	$5^2{13}^4{59}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$13\cdot 5^2,𝒦$
686	$769$		1b	$5^2{769}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
687	*$770$	$2\cdot 5\cdot 7\cdot 11$	1a	$5^6{2}^4{7}^4{11}^4$	$64$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$2\cdot 5,2^2\cdot 11$
688	$771$	$3\cdot 257$	1b	$5^2{3}^4{257}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
689	$772$	$2^2\cdot 193$	1b	$5^2{2}^4{193}^4$	$4$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$2$
690	$773$		1b	$5^2{773}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
691	$774$	$2\cdot 3^2\cdot 43$	2	$2^4{3}^4{43}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
692	$775$	$5^2\cdot 31$	1a	$5^6{31}^4$	$4$	$2$	$3$	$5$	$2$	$(-,-,\otimes ,-)$	${\alpha }_1$	${𝔎}_1,{𝔎}_2$
693	$776$	$2^3\cdot 97$	2	$2^4{97}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
694	$777$	$3\cdot 7\cdot 37$	1b	$5^2{3}^4{7}^4{37}^4$	$12$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$3,7\cdot 5$
695	$778$	$2\cdot 389$	1b	$5^2{2}^4{389}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$5,𝒦$
696	*$779$	$19\cdot 41$	1b	$5^2{19}^4{41}^4$	$3$	$3$	$4$	$8$	$1$	$(-,\otimes ,\otimes ,-)$	${\alpha }_2$	${𝒦}_{\left(19\right)}{𝒦}_{\left(41\right)},$
												${𝔎}_{\left(41\right),1}$
697	$780$	$2^2\cdot 3\cdot 5\cdot 13$	1a	$5^6{2}^4{3}^4{13}^4$	$64$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2\cdot 5,5\cdot 13$
698	$781$	$11\cdot 71$	1b	$5^2{11}^4{71}^4$	$3$	$3$	$5$	$9$	$2$	$(-,-,\otimes ,-)$	${\alpha }_1$	${𝔎}_{\left(11\right),1}{𝔎}_{\left(71\right),1}^4,$
												${𝔎}_{\left(11\right),2}{𝔎}_{\left(71\right),2}^2$
699	$782$	$2\cdot 17\cdot 23$	2	$2^4{17}^4{23}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
700	$783$	$3^3\cdot 29$	1b	$5^2{3}^4{29}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$29,ℒ$
701	*$785$	$5\cdot 157$	1a	$5^6{157}^4$	$4$	$1$	$2$	$4$	$5$	$(-,-,-,\times )$	$\epsilon$
702	$786$	$2\cdot 3\cdot 131$	1b	$5^2{2}^4{3}^4{131}^4$	$13$	$2$	$3$	$7$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$2\cdot 3\cdot 5^2,𝒦$
703	$787$		1b	$5^2{787}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
704	$788$	$2^2\cdot 197$	1b	$5^2{2}^4{197}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
705	$789$	$3\cdot 263$	1b	$5^2{3}^4{263}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
706	$790$	$2\cdot 5\cdot 79$	1a	$5^6{2}^4{79}^4$	$16$	$1$	$2$	$4$	$5$	$(-,\times ,-,-)$	$\epsilon$	$5$
707	$791$	$7\cdot 113$	1b	$5^2{7}^4{113}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
708	$792$	$2^3\cdot 3^2\cdot 11$	1b	$5^2{2}^4{3}^4{11}^4$	$13$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$2\cdot 5,11\cdot 5$
709	$793$	$13\cdot 61$	2	${13}^4{61}^4$	$1$	$1$	$2$	$4$	$5$	$(-,-,\otimes ,-)$	${\beta }_1$	$61,{𝔎}_2$
710	$794$	$2\cdot 397$	1b	$5^2{2}^4{397}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
711	$795$	$3\cdot 5\cdot 53$	1a	$5^6{3}^4{53}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
712	$796$	$2^2\cdot 199$	1b	$5^2{2}^4{199}^4$	$4$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$2\cdot 5,𝒦$
713	$797$		1b	$5^2{797}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
714	*$798$	$2\cdot 3\cdot 7\cdot 19$	1b	$5^2{2}^4{3}^4{7}^4{19}^4$	$52$	$2$	$4$	$9$	$6$	$(-,\times ,-,-)$	$\gamma$	$3\cdot 19,7\cdot 5^4$
715	$799$	$17\cdot 47$	2	${17}^4{47}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$

Table 22. Table 22. 46 46 46 pure metacyclic fields with normalized radicands 801 ≤ D ≤ 848 801 𝐷 848 801\leq D\leq 848

No.	$D$	Factors	S	$f^4$	$m$	$V_L$	$V_M$	$V_N$	$E$	$(1,2,4,5)$	T	P
716	$801$	$3^2\cdot 89$	2	$3^4{89}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$𝒦$
717	$802$	$2\cdot 401$	1b	$5^2{2}^4{401}^4$	$4$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$2^2\cdot 5,𝒦$
718	$803$	$11\cdot 73$	1b	$5^2{11}^4{73}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$73\cdot 5^2,𝒦$
719	$804$	$2^2\cdot 3\cdot 67$	1b	$5^2{2}^4{3}^4{67}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$3,2\cdot 5^3$
720	$805$	$5\cdot 7\cdot 23$	1a	$5^6{7}^4{23}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
721	$806$	$2\cdot 13\cdot 31$	1b	$5^2{2}^4{13}^4{31}^4$	$13$	$2$	$3$	$6$	$3$	$(-,-,\otimes ,-)$	${\beta }_1$	$31\cdot 5^2,{𝔎}_2$
722	$807$	$3\cdot 269$	2	$3^4{269}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$𝒦$
723	*$808$	$2^3\cdot 101$	1b	$5^2{2}^4{101}^4$	$4$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
724	$809$		1b	$5^2{809}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
725	$810$	$2\cdot 3^4\cdot 5$	1a	$5^6{2}^4{3}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
726	$811$		1b	$5^2{811}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$ℒ,{𝔎}_1$
727	$812$	$2^2\cdot 7\cdot 29$	1b	$5^2{2}^4{7}^4{29}^4$	$12$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$29,7^2\cdot 5$
728	$813$	$3\cdot 271$	1b	$5^2{3}^4{271}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$5,𝒦$
729	$814$	$2\cdot 11\cdot 37$	1b	$5^2{2}^4{11}^4{37}^4$	$13$	$2$	$3$	$6$	$3$	$(-,-,\otimes ,-)$	${\beta }_1$	$11\cdot {37}^2\cdot 5,{𝔎}_2$
730	$815$	$5\cdot 163$	1a	$5^6{163}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
731	$816$	$2^4\cdot 3\cdot 17$	1b	$5^2{2}^4{3}^4{17}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2\cdot 5^2,3\cdot 5^3$
732	$817$	$19\cdot 43$	1b	$5^2{19}^4{43}^4$	$4$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	${19}^2\cdot 5,𝒦$
733	$818$	$2\cdot 409$	2	$2^4{409}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$𝒦$
734	$819$	$3^2\cdot 7\cdot 13$	1b	$5^2{3}^4{7}^4{13}^4$	$12$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$13,7\cdot 5^2$
735	$820$	$2^2\cdot 5\cdot 41$	1a	$5^6{2}^4{41}^4$	$16$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
736	$821$		1b	$5^2{821}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$ℒ,{𝔎}_2$
737	$822$	$2\cdot 3\cdot 137$	1b	$5^2{2}^4{3}^4{137}^4$	$13$	$2$	$4$	$8$	$5$	$(\times ,-,-,-)$	$\epsilon$	$3^3\cdot 137\cdot 5$
738	$823$		1b	$5^2{823}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
739	$824$	$2^3\cdot 103$	2	$2^4{103}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
740	*$825$	$3\cdot 5^2\cdot 11$	1a	$5^6{3}^4{11}^4$	$16$	$1$	$2$	$4$	$5$	$(-,-,\otimes ,-)$	${\beta }_1$	$3^2\cdot 11,{𝔎}_1$
741	$826$	$2\cdot 7\cdot 59$	2	$2^4{7}^4{59}^4$	$4$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$59,ℒ$
742	$827$		1b	$5^2{827}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
743	$828$	$2^2\cdot 3^2\cdot 23$	1b	$5^2{2}^4{3}^4{23}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$23,2\cdot 5^2$
744	$829$		1b	$5^2{829}^4$	$1$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$829,ℒ$
745	$830$	$2\cdot 5\cdot 83$	1a	$5^6{2}^4{83}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
746	$831$	$3\cdot 277$	1b	$5^2{3}^4{277}^4$	$3$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$3^2\cdot 5$
747	$833$	$7^2\cdot 17$	1b	$5^2{7}^4{17}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
748	$834$	$2\cdot 3\cdot 139$	1b	$5^2{2}^4{3}^4{139}^4$	$13$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$2\cdot 5,3\cdot 5^3$
749	$835$	$5\cdot 167$	1a	$5^6{167}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
750	$836$	$2^2\cdot 11\cdot 19$	1b	$5^2{2}^4{11}^4{19}^4$	$13$	$2$	$3$	$7$	$4$	$(-,\otimes ,(\times ),-)$	${\beta }_2$	$2\cdot 11\cdot 5,{𝒦}_{\left(11\right)}\cdot {𝒦}_{\left(19\right)}^3$
751	$837$	$3^3\cdot 31$	1b	$5^2{3}^4{31}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$5,𝒦$
752	$838$	$2\cdot 419$	1b	$5^2{2}^4{419}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$2\cdot 5,𝒦$
753	$839$		1b	$5^2{839}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
754	$840$	$2^3\cdot 3\cdot 5\cdot 7$	1a	$5^6{2}^4{3}^4{7}^4$	$64$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2\cdot 7,3\cdot 7$
755	$842$	$2\cdot 421$	1b	$5^2{2}^4{421}^4$	$3$	$1$	$2$	$4$	$5$	$(-,-,\times ,-)$	$\epsilon$	$2\cdot 5^3$
756	*$843$	$3\cdot 281$	2	$3^4{281}^4$	$1$	$1$	$2$	$3$	$4$	$(-,-,\otimes ,-)$	${\delta }_1$	${𝔎}_1$
757	$844$	$2^2\cdot 211$	1b	$5^2{2}^4{211}^4$	$3$	$1$	$2$	$4$	$5$	$(-,-,\times ,-)$	$\epsilon$	$2\cdot 5^2$
758	$845$	$5\cdot {13}^2$	1a	$5^6{13}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
759	$846$	$2\cdot 3^2\cdot 47$	1b	$5^2{2}^4{3}^4{47}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2,3$
760	$847$	$7\cdot {11}^2$	1b	$5^2{7}^4{11}^4$	$4$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$7\cdot 5^2,𝒦$
761	$848$	$2^4\cdot 53$	1b	$5^2{2}^4{53}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$

Table 23. Table 23. 47 47 47 pure metacyclic fields with normalized radicands 849 ≤ D ≤ 901 849 𝐷 901 849\leq D\leq 901

No.	$D$	Factors	S	$f^4$	$m$	$V_L$	$V_M$	$V_N$	$E$	$(1,2,4,5)$	T	P
762	$849$	$3\cdot 283$	2	$3^4{283}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
763	$850$	$2\cdot 5^2\cdot 17$	1a	$5^6{2}^4{17}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
764	$851$	$23\cdot 37$	2	${23}^4{37}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
765	$852$	$2^2\cdot 3\cdot 71$	1b	$5^2{2}^4{3}^4{71}^4$	$13$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$2\cdot 5^3,3\cdot 5^3$
766	$853$		1b	$5^2{853}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
767	$854$	$2\cdot 7\cdot 61$	1b	$5^2{2}^4{7}^4{61}^4$	$12$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$2\cdot 5^2,61$
768	$855$	$3^2\cdot 5\cdot 19$	1a	$5^6{3}^4{19}^4$	$16$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$3,𝒦$
769	$856$	$2^3\cdot 107$	1b	$5^2{2}^4{107}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
770	$857$		2	${857}^4$	$1$	$0$	$0$	$0$	$5$	$(-,-,-,\otimes )$	$\vartheta$
771	*$858$	$2\cdot 3\cdot 11\cdot 13$	1b	$5^2{2}^4{3}^4{11}^4{13}^4$	$51$	$2$	$4$	$9$	$6$	$(-,-,\times ,-)$	$\gamma$	$2\cdot 5,13\cdot 5$
772	$859$		1b	$5^2{859}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
773	$860$	$2^2\cdot 5\cdot 43$	1a	$5^6{2}^4{43}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
774	*$861$	$3\cdot 7\cdot 41$	1b	$5^2{3}^4{7}^4{41}^4$	$12$	$3$	$4$	$8$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
775	$862$	$2\cdot 431$	1b	$5^2{2}^4{431}^4$	$3$	$1$	$2$	$4$	$5$	$(-,-,\otimes ,-)$	${\beta }_1$	$431,{𝔎}_1$
776	$863$		1b	$5^2{863}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
777	$865$	$5\cdot 173$	1a	$5^6{173}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
778	$866$	$2\cdot 433$	1b	$5^2{2}^4{433}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
779	$868$	$2^2\cdot 7\cdot 31$	2	$2^4{7}^4{31}^4$	$4$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$2\cdot 7^2,𝒦$
780	$869$	$11\cdot 79$	1b	$5^2{11}^4{79}^4$	$3$	$2$	$2$	$5$	$2$	$(-,\otimes ,(\times ),-)$	${\alpha }_3$	${𝒦}_{\left(11\right)},{𝒦}_{\left(79\right)}$
781	$870$	$2\cdot 3\cdot 5\cdot 29$	1a	$5^6{2}^4{3}^4{29}^4$	$64$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$29,3^2\cdot 5$
782	$871$	$13\cdot 67$	1b	$5^2{13}^4{67}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
783	$872$	$2^3\cdot 109$	1b	$5^2{2}^4{109}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$109,ℒ$
784	$873$	$3^2\cdot 97$	1b	$5^2{3}^4{97}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
785	$874$	$2\cdot 19\cdot 23$	2	$2^4{19}^4{23}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$2\cdot {19}^2,𝒦$
786	$876$	$2^2\cdot 3\cdot 73$	2	$2^4{3}^4{73}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
787	$877$		1b	$5^2{877}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
788	$878$	$2\cdot 439$	1b	$5^2{2}^4{439}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$5,𝒦$
789	$879$	$3\cdot 293$	1b	$5^2{3}^4{293}^4$	$4$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$3$
790	$880$	$2^4\cdot 5\cdot 11$	1a	$5^6{2}^4{11}^4$	$16$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_2$
791	$881$		1b	$5^2{881}^4$	$1$	$1$	$2$	$3$	$4$	$(-,-,\otimes ,-)$	${\delta }_1$	${𝔎}_2$
792	$883$		1b	$5^2{883}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
793	$884$	$2^2\cdot 13\cdot 17$	1b	$5^2{2}^4{13}^4{17}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$17,13\cdot 5^2$
794	$885$	$3\cdot 5\cdot 59$	1a	$5^6{3}^4{59}^4$	$16$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$3,𝒦$
795	$886$	$2\cdot 443$	1b	$5^2{2}^4{443}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
796	$887$		1b	$5^2{887}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
797	$888$	$2^3\cdot 3\cdot 37$	1b	$5^2{2}^4{3}^4{37}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2,3$
798	$889$	$7\cdot 127$	1b	$5^2{7}^4{127}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
799	$890$	$2\cdot 5\cdot 89$	1a	$5^6{2}^4{89}^4$	$16$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$2^2\cdot 5,𝒦$
800	$891$	$3^4\cdot 11$	1b	$5^2{3}^4{11}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$3\cdot 5^3,𝒦$
801	$892$	$2^2\cdot 223$	1b	$5^2{2}^4{223}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
802	*$893$	$19\cdot 47$	2	${19}^4{47}^4$	$1$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$19,𝔏$
803	*$894$	$2\cdot 3\cdot 149$	1b	$5^2{2}^4{3}^4{149}^4$	$12$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$5,2\cdot 3^2$
804	$895$	$5\cdot 179$	1a	$5^6{179}^4$	$4$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$𝒦$
805	$897$	$3\cdot 13\cdot 23$	1b	$5^2{3}^4{13}^4{23}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$13\cdot 5^2,3^3\cdot 5^2$
806	$898$	$2\cdot 449$	1b	$5^2{2}^4{449}^4$	$4$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$2\cdot 5^2,𝒦$
807	$899$	$29\cdot 31$	2	${29}^4{31}^4$	$1$	$2$	$2$	$5$	$2$	$(-,\otimes ,(\times ),-)$	${\alpha }_3$	${𝒦}_{\left(29\right)},{𝒦}_{\left(31\right)}$
808	$901$	$17\cdot 53$	2	${17}^4{53}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$

Table 24. Table 24. 47 47 47 pure metacyclic fields with normalized radicands 902 ≤ D ≤ 949 902 𝐷 949 902\leq D\leq 949

No.	$D$	Factors	S	$f^4$	$m$	$V_L$	$V_M$	$V_N$	$E$	$(1,2,4,5)$	T	P
809	*$902$	$2\cdot 11\cdot 41$	1b	$5^2{2}^4{11}^4{41}^4$	$13$	$2$	$3$	$7$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$11\cdot 5,{𝒦}_{\left(11\right)}\cdot {𝒦}_{\left(41\right)}$
810	$903$	$3\cdot 7\cdot 43$	1b	$5^2{3}^4{7}^4{43}^4$	$16$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$5,7^2\cdot 43$
811	$904$	$2^3\cdot 113$	1b	$5^2{2}^4{113}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
812	$905$	$5\cdot 181$	1a	$5^6{181}^4$	$4$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
813	$906$	$2\cdot 3\cdot 151$	1b	$5^2{2}^4{3}^4{151}^4$	$12$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$2,3\cdot 5^2$
814	$907$		2	${907}^4$	$1$	$0$	$0$	$0$	$5$	$(-,-,-,\otimes )$	$\vartheta$
815	$908$	$2^2\cdot 227$	1b	$5^2{2}^4{227}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
816	$909$	$3^2\cdot 101$	1b	$5^2{3}^4{101}^4$	$4$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
817	$910$	$2\cdot 5\cdot 7\cdot 13$	1a	$5^6{2}^4{7}^4{13}^4$	$64$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2,13\cdot 5^2$
818	$911$		1b	$5^2{911}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$ℒ,{𝔎}_2$
819	$912$	$2^4\cdot 3\cdot 19$	1b	$5^2{2}^4{3}^4{19}^4$	$13$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$3,2\cdot 5^2$
820	$913$	$11\cdot 83$	1b	$5^2{11}^4{83}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$11\cdot 5,𝒦$
821	$914$	$2\cdot 457$	1b	$5^2{2}^4{457}^4$	$4$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$2$
822	$915$	$3\cdot 5\cdot 61$	1a	$5^6{3}^4{61}^4$	$16$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$3\cdot 5^2,𝒦$
823	$916$	$2^2\cdot 229$	1b	$5^2{2}^4{229}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$2^4\cdot 5,𝒦$
824	$917$	$7\cdot 131$	1b	$5^2{7}^4{131}^4$	$4$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$131\cdot 5,𝒦$
825	$918$	$2\cdot 3^3\cdot 17$	2	$2^4{3}^4{17}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
826	$919$		1b	$5^2{919}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
827	$920$	$2^3\cdot 5\cdot 23$	1a	$5^6{2}^4{23}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
828	$921$	$3\cdot 307$	1b	$5^2{3}^4{307}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
829	$922$	$2\cdot 461$	1b	$5^2{2}^4{461}^4$	$3$	$1$	$2$	$4$	$5$	$(-,-,\times ,-)$	$\epsilon$	$2^2\cdot 5$
830	$923$	$13\cdot 71$	1b	$5^2{13}^4{71}^4$	$3$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_2$
831	*$924$	$2^2\cdot 3\cdot 7\cdot 11$	2	$2^4{3}^4{7}^4{11}^4$	$12$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$3\cdot 7,7\cdot 11$
832	$925$	$5^2\cdot 37$	1a	$5^6{37}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
833	$926$	$2\cdot 463$	2	$2^4{463}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
834	$927$	$3^2\cdot 103$	1b	$5^2{3}^4{103}^4$	$3$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$3^3\cdot 5^2$
835	$929$		1b	$5^2{929}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
836	$930$	$2\cdot 3\cdot 5\cdot 31$	1a	$5^6{2}^4{3}^4{31}^4$	$64$	$2$	$4$	$9$	$6$	$(-,-,\times ,-)$	$\gamma$	$5,31$
837	$931$	$7^2\cdot 19$	1b	$5^2{7}^4{19}^4$	$4$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$7,ℒ$
838	$932$	$2^2\cdot 233$	2	$2^4{233}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
839	$933$	$3\cdot 311$	1b	$5^2{3}^4{311}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$3\cdot 5,𝒦$
840	$934$	$2\cdot 467$	1b	$5^2{2}^4{467}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
841	$935$	$5\cdot 11\cdot 17$	1a	$5^6{11}^4{17}^4$	$16$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_2$
842	$936$	$2^3\cdot 3^2\cdot 13$	1b	$5^2{2}^4{3}^4{13}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$3,5$
843	$937$		1b	$5^2{937}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
844	$938$	$2\cdot 7\cdot 67$	1b	$5^2{2}^4{7}^4{67}^4$	$12$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$7,67\cdot 5$
845	$939$	$3\cdot 313$	1b	$5^2{3}^4{313}^4$	$3$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$5$
846	$940$	$2^2\cdot 5\cdot 47$	1a	$5^6{2}^4{47}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
847	$941$		1b	$5^2{941}^4$	$1$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$ℒ,{𝔎}_1$
848	$942$	$2\cdot 3\cdot 157$	1b	$5^2{2}^4{3}^4{157}^4$	$12$	$2$	$4$	$8$	$5$	$(\times ,-,-,-)$	$\epsilon$	$157$
849	$943$	$23\cdot 41$	2	${23}^4{41}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_2$
850	$944$	$2^4\cdot 59$	1b	$5^2{2}^4{59}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$2\cdot 5^4,𝒦$
851	$945$	$3^3\cdot 5\cdot 7$	1a	$5^6{3}^4{7}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
852	$946$	$2\cdot 11\cdot 43$	1b	$5^2{2}^4{11}^4{43}^4$	$12$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$2\cdot 5,43$
853	$947$		1b	$5^2{947}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
854	$948$	$2^2\cdot 3\cdot 79$	1b	$5^2{2}^4{3}^4{79}^4$	$13$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$2\cdot 5^2,3\cdot 5^2$
855	$949$	$13\cdot 73$	2	${13}^4{73}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$

Table 25. Table 25. 45 45 45 pure metacyclic fields with normalized radicands 950 ≤ D ≤ 999 950 𝐷 999 950\leq D\leq 999

No.	$D$	Factors	S	$f^4$	$m$	$V_L$	$V_M$	$V_N$	$E$	$(1,2,4,5)$	T	P
856	$950$	$2\cdot 5^2\cdot 19$	1a	$5^6{2}^4{19}^4$	$16$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$5,𝒦$
857	$951$	$3\cdot 317$	2	$3^4{317}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
858	$952$	$2^3\cdot 7\cdot 17$	1b	$5^2{2}^4{7}^4{17}^4$	$12$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2\cdot 5^4,7\cdot 5^4$
859	$953$		1b	$5^2{953}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
860	$954$	$2\cdot 3^2\cdot 53$	1b	$5^2{2}^4{3}^4{53}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$3\cdot 5,2\cdot 5^2$
861	*$955$	$5\cdot 191$	1a	$5^6{191}^4$	$4$	$2\ast$	$3\ast$	$6\ast$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
862	$956$	$2^2\cdot 239$	1b	$5^2{2}^4{239}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$239,ℒ$
863	*$957$	$3\cdot 11\cdot 29$	2	$3^4{11}^4{29}^4$	$3$	$2$	$2$	$5$	$2$	$(-,\otimes ,(\times ),-)$	${\alpha }_3$	${𝒦}_{\left(11\right)},{𝒦}_{\left(29\right)}$
864	$958$	$2\cdot 479$	1b	$5^2{2}^4{479}^4$	$3$	$1$	$2$	$4$	$5$	$(-,\times ,-,-)$	$\epsilon$	$2^3\cdot 5$
865	$959$	$7\cdot 137$	1b	$5^2{7}^4{137}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
866	$962$	$2\cdot 13\cdot 37$	1b	$5^2{2}^4{13}^4{37}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2\cdot 5,37\cdot 5$
867	$963$	$3^2\cdot 107$	1b	$5^2{3}^4{107}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
868	$964$	$2^2\cdot 241$	1b	$5^2{2}^4{241}^4$	$3$	$1$	$2$	$4$	$5$	$(-,-,\otimes ,-)$	${\beta }_1$	$241,{𝔎}_2$
869	$965$	$5\cdot 193$	1a	$5^6{193}^4$	$4$	$0$	$0$	$1$	$6$	$(-,-,-,\otimes )$	$\eta$
870	$966$	$2\cdot 3\cdot 7\cdot 23$	1b	$5^2{2}^4{3}^4{7}^4{23}^4$	$52$	$2$	$4$	$9$	$6$	$(\times ,-,-,-)$	$\gamma$	$3\cdot 5^3,2\cdot {23}^2$
871	$967$		1b	$5^2{967}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
872	$969$	$3\cdot 17\cdot 19$	1b	$5^2{3}^4{17}^4{19}^4$	$13$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$19,3\cdot 5^2$
873	$970$	$2\cdot 5\cdot 97$	1a	$5^6{2}^4{97}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
874	$971$		1b	$5^2{971}^4$	$1$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$ℒ,{𝔎}_1$
875	$973$	$7\cdot 139$	1b	$5^2{7}^4{139}^4$	$4$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$7\cdot 5^2,𝒦$
876	$974$	$2\cdot 487$	2	$2^4{487}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
877	$975$	$3\cdot 5^2\cdot 13$	1a	$5^6{3}^4{13}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
878	$976$	$2^4\cdot 61$	2	$2^4{61}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_2$
879	$977$		1b	$5^2{977}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
880	$978$	$2\cdot 3\cdot 163$	1b	$5^2{2}^4{3}^4{163}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$3\cdot 5^2,2\cdot 5^4$
881	$979$	$11\cdot 89$	1b	$5^2{11}^4{89}^4$	$3$	$2$	$3$	$7$	$4$	$(-,\otimes ,(\times ),-)$	${\beta }_2$	$11,{𝒦}_{\left(11\right)}\cdot {𝒦}_{\left(89\right)}^2$
882	$980$	$2^2\cdot 5\cdot 7^2$	1a	$5^6{2}^4{7}^4$	$16$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$7$
883	$981$	$3^2\cdot 109$	1b	$5^2{3}^4{109}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$109,ℒ$
884	*$982$	$2\cdot 491$	2	$2^4{491}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
885	$983$		1b	$5^2{983}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
886	$984$	$2^3\cdot 3\cdot 41$	1b	$5^2{2}^4{3}^4{41}^4$	$13$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$3,2\cdot 5^2$
887	$985$	$5\cdot 197$	1a	$5^6{197}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
888	$986$	$2\cdot 17\cdot 29$	1b	$5^2{2}^4{17}^4{29}^4$	$13$	$2$	$4$	$8$	$5$	$(-,\times ,-,-)$	$\epsilon$	$2\cdot 5$
889	$987$	$3\cdot 7\cdot 47$	1b	$5^2{3}^4{7}^4{47}^4$	$12$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$7\cdot 5^2,3\cdot 5^4$
890	$988$	$2^2\cdot 13\cdot 19$	1b	$5^2{2}^4{13}^4{19}^4$	$13$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$2,13\cdot 5^2$
891	$989$	$23\cdot 43$	1b	$5^2{23}^4{43}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
892	$990$	$2\cdot 3^2\cdot 5\cdot 11$	1a	$5^6{2}^4{3}^4{11}^4$	$64$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$2\cdot 5,3\cdot 5^2$
893	$991$		1b	$5^2{991}^4$	$1$	$1$	$2$	$3$	$4$	$(-,-,\otimes ,-)$	${\delta }_1$	${𝔎}_1$
894	$993$	$3\cdot 331$	2	$3^4{331}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_2$
895	$994$	$2\cdot 7\cdot 71$	1b	$5^2{2}^4{7}^4{71}^4$	$12$	$2$	$3$	$7$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$2\cdot 5^3,𝒦$
896	$995$	$5\cdot 199$	1a	$5^6{199}^4$	$4$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,\otimes )$	${\zeta }_2$	$𝒦$
897	$996$	$2^2\cdot 3\cdot 83$	1b	$5^2{2}^4{3}^4{83}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2\cdot 5^2,3\cdot 5^4$
898	$997$		1b	$5^2{997}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
899	$998$	$2\cdot 499$	1b	$5^2{2}^4{499}^4$	$4$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$499,ℒ$
900	$999$	$3^3\cdot 37$	2	$3^4{37}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$

Table 26. Table 26. Absolute frequencies of differential principal factorization types

Type	$100$	$200$	$300$	$400$	$500$	$600$	$700$	$800$	$900$	$1000$	$\%$
${\alpha }_1$	$1$	$2$	$3$	$4$	$5$	$5$	$5$	$9$	$9$	$9$
${\alpha }_2$	$10$	$17$	$23$	$30$	$35$	$42$	$52$	$57$	$63$	$75$	$8.3$
${\alpha }_3$	$0$	$0$	$0$	$1$	$1$	$3$	$5$	$5$	$7$	$8$
${\beta }_1$	$0$	$2$	$4$	$7$	$8$	$11$	$15$	$18$	$22$	$23$
${\beta }_2$	$7$	$24$	$40$	$54$	$80$	$94$	$108$	$126$	$146$	$161$	$17.9$
$\gamma$	$25$	$55$	$88$	$117$	$148$	$187$	$222$	$259$	$290$	$324$	$36.0$
${\delta }_1$	$0$	$0$	$1$	$1$	$3$	$4$	$4$	$4$	$6$	$7$
${\delta }_2$	$8$	$14$	$19$	$23$	$31$	$35$	$38$	$44$	$51$	$53$	$5.9$
$\epsilon$	$26$	$45$	$67$	$95$	$110$	$128$	$150$	$165$	$184$	$208$	$23.1$
${\zeta }_1$	$0$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
${\zeta }_2$	$0$	$0$	$0$	$0$	$0$	$1$	$1$	$4$	$4$	$5$
$\eta$	$1$	$2$	$4$	$5$	$5$	$6$	$6$	$6$	$6$	$7$
$\vartheta$	$3$	$6$	$8$	$9$	$11$	$13$	$15$	$17$	$18$	$19$
Total	$81$	$168$	$258$	$347$	$438$	$530$	$622$	$715$	$807$	$900$	$100.0$

Table 27. Table 27. 46 46 46 prototypes 2 ≤ 𝐌 < 200 2 𝐌 200 2\leq\mathbf{M}<200 of pure metacyclic fields

No.	$𝐌$	Factors	S	$f^4$	$m$	$V_L$	$V_M$	$V_N$	$E$	$(1,2,4,5)$	T	P	$\left|𝐌\right|$
1	$2$		1b	$5^2{2}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$		$71$
2	$5$		1a	$5^6$	$1$	$0$	$0$	$0$	$5$	$(-,-,-,\otimes )$	$\vartheta$		$1$
3	$6$	$2\cdot 3$	1b	$5^2{2}^4{3}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$		$77$
4	$7$		2	$7^4$	$1$	$0$	$0$	$0$	$5$	$(-,-,-,\otimes )$	$\vartheta$		$18$
5	$10$	$2\cdot 5$	1a	$5^6{2}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$		$31$
6	$11$		1b	$5^2{11}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$ℒ,{𝔎}_1$	$14$
7	$14$	$2\cdot 7$	1b	$5^2{2}^4{7}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$		$44$
8	$18$	$2\cdot 3^2$	2	$2^4{3}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$		$37$
9	$19$		1b	$5^2{19}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$	$27$
10	$22$	$2\cdot 11$	1b	$5^2{2}^4{11}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$2\cdot 5,𝒦$	$35$
11	$30$	$2\cdot 3\cdot 5$	1a	$5^6{2}^4{3}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$		$37$
12	$31$		1b	$5^2{31}^4$	$1$	$2$	$3$	$5$	$2$	$(-,-,\otimes ,-)$	${\alpha }_1$	${𝔎}_1,{𝔎}_2$	$2$
13	$33$	$3\cdot 11$	1b	$5^2{3}^4{11}^4$	$3$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_2$	$8$
14	$35$	$5\cdot 7$	1a	$5^6{7}^4$	$4$	$0$	$0$	$1$	$6$	$(-,-,-,\otimes )$	$\eta$		$6$
15	$38$	$2\cdot 19$	1b	$5^2{2}^4{19}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$5,𝒦$	$44$
16	$42$	$2\cdot 3\cdot 7$	1b	$5^2{2}^4{3}^4{7}^4$	$12$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2\cdot 5,3\cdot 5^2$	$22$
17	$55$	$5\cdot 11$	1a	$5^6{11}^4$	$4$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$	$6$
18	$57$	$3\cdot 19$	2	$3^4{19}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$𝒦$	$10$
19	$66$	$2\cdot 3\cdot 11$	1b	$5^2{2}^4{3}^4{11}^4$	$13$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$2\cdot 5,3\cdot 5^3$	$17$
20	$70$	$2\cdot 5\cdot 7$	1a	$5^6{2}^4{7}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$		$14$
21	$77$	$7\cdot 11$	1b	$5^2{7}^4{11}^4$	$4$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$11\cdot 5^3,𝒦$	$7$
22	$78$	$2\cdot 3\cdot 13$	1b	$5^2{2}^4{3}^4{13}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$3,2\cdot 5^3$	$37$
23	$82$	$2\cdot 41$	2	$2^4{41}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_2$	$15$
24	$95$	$5\cdot 19$	1a	$5^6{19}^4$	$4$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$𝒦$	$9$
25	$101$		2	${101}^4$	$1$	$1$	$2$	$4$	$5$	$(-,-,\otimes ,\otimes )$	${\zeta }_1$	${𝔎}_2$	$1$
26	$110$	$2\cdot 5\cdot 11$	1a	$5^6{2}^4{11}^4$	$16$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$11,ℒ$	$11$
27	$114$	$2\cdot 3\cdot 19$	1b	$5^2{2}^4{3}^4{19}^4$	$13$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$2\cdot 5^3,3\cdot 5^3$	$20$
28	$123$	$3\cdot 41$	1b	$5^2{3}^4{41}^4$	$3$	$2$	$3$	$6$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_2$	$2$
29	$126$	$2\cdot 3^2\cdot 7$	2	$2^4{3}^4{7}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$		$6$
30	$131$		1b	$5^2{131}^4$	$1$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$ℒ,{𝔎}_2$	$6$
31	$132$	$2^2\cdot 3\cdot 11$	2	$2^4{3}^4{11}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$11,ℒ$	$5$
32	$133$	$7\cdot 19$	1b	$5^2{7}^4{19}^4$	$4$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$7,ℒ$	$7$
33	$139$		1b	$5^2{139}^4$	$1$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$5,ℒ$	$4$
34	$140$	$2^2\cdot 5\cdot 7$	1a	$5^6{2}^4{7}^4$	$16$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$7$	$5$
35	$141$	$3\cdot 47$	1b	$5^2{3}^4{47}^4$	$3$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$47\cdot 5$	$19$
36	$149$		2	${149}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,\times )$	${\delta }_2$	$ℒ$	$6$
37	$151$		2	${151}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,\times )$	${\alpha }_2$	${𝔎}_1$	$3$
38	$154$	$2\cdot 7\cdot 11$	1b	$5^2{2}^4{7}^4{11}^4$	$12$	$2$	$3$	$7$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$2\cdot {11}^2,𝒦$	$3$
39	$155$	$5\cdot 31$	1a	$5^6{31}^4$	$4$	$2$	$3$	$5$	$2$	$(-,-,\otimes ,-)$	${\alpha }_1$	${𝔎}_1,{𝔎}_2$	$2$
40	$171$	$3^2\cdot 19$	1b	$5^2{3}^4{19}^4$	$3$	$1$	$2$	$4$	$5$	$(-,\times ,-,-)$	$\epsilon$	$19\cdot 5^2$	$6$
41	$174$	$2\cdot 3\cdot 29$	2	$2^4{3}^4{29}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$3^2\cdot 29,𝒦$	$3$
42	$180$	$2^2\cdot 3^2\cdot 5$	1a	$5^6{2}^4{3}^4$	$16$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$3$	$5$
43	$182$	$2\cdot 7\cdot 13$	2	$2^4{7}^4{13}^4$	$4$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$7$	$1$
44	$186$	$2\cdot 3\cdot 31$	1b	$5^2{2}^4{3}^4{31}^4$	$13$	$2$	$3$	$6$	$3$	$(-,-,\otimes ,-)$	${\beta }_1$	$5,{𝔎}_2$	$7$
45	$190$	$2\cdot 5\cdot 19$	1a	$5^6{2}^4{19}^4$	$16$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$5,𝒦$	$9$
46	$191$		1b	$5^2{191}^4$	$1$	$1$	$2$	$4$	$5$	$(-,-,\otimes ,-)$	${\beta }_1$	$5,{𝔎}_1$	$3$

Table 28. Table 28. 44 44 44 prototypes 200 < 𝐌 < 510 200 𝐌 510 200<\mathbf{M}<510 of pure metacyclic fields

No.	$𝐌$	Factors	S	$f^4$	$m$	$V_L$	$V_M$	$V_N$	$E$	$(1,2,4,5)$	T	P	$\left|𝐌\right|$
47	$202$	$2\cdot 101$	1b	$5^2{2}^4{101}^4$	$4$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$2\cdot 5,𝒦$	$6$
48	$203$	$7\cdot 29$	1b	$5^2{7}^4{29}^4$	$4$	$1$	$2$	$4$	$5$	$(-,\times ,-,-)$	$\epsilon$	$29\cdot 5$	$2$
49	$209$	$11\cdot 19$	1b	$5^2{11}^4{19}^4$	$3$	$2$	$3$	$7$	$4$	$(-,\otimes ,\times ,-)$	${\beta }_2$	$11\cdot 5^2,{𝒦}_{\left(19\right)}$	$2$
50	$210$	$2\cdot 3\cdot 5\cdot 7$	1a	$5^6{2}^4{3}^4{7}^4$	$64$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$7,3^2\cdot 5$	$5$
51	$211$		1b	$5^2{211}^4$	$1$	$3$	$5$	$9$	$2$	$(-,-,\otimes ,-)$	${\delta }_1$	${𝔎}_2$	$1$
52	$218$	$2\cdot 109$	2	$2^4{109}^4$	$1$	$1$	$2$	$4$	$5$	$(-,\times ,-,-)$	$\epsilon$	$2$	$1$
53	$231$	$3\cdot 7\cdot 11$	1b	$5^2{3}^4{7}^4{11}^4$	$12$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$11,7\cdot 5^2$	$5$
54	$247$	$13\cdot 19$	1b	$5^2{13}^4{19}^4$	$3$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$		$2$
55	$253$	$11\cdot 23$	1b	$5^2{11}^4{23}^4$	$3$	$1$	$2$	$4$	$5$	$(-,-,\otimes ,-)$	${\beta }_1$	$11,{𝔎}_1$	$4$
56	$259$	$7\cdot 37$	1b	$5^2{7}^4{37}^4$	$4$	$1$	$2$	$5\ast$	$6$	$(\times ,-,-,-)$	$\gamma$		$1$
57	$266$	$2\cdot 7\cdot 19$	1b	$5^2{2}^4{7}^4{19}^4$	$12$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$5,2^3\cdot 7$	$3$
58	$273$	$3\cdot 7\cdot 13$	1b	$5^2{3}^4{7}^4{13}^4$	$12$	$2$	$4$	$8$	$5$	$(\times ,-,-,-)$	$\epsilon$	$7\cdot {13}^2\cdot 5^4$	$4$
59	$275$	$5^2\cdot 11$	1a	$5^6{11}^4$	$4$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_2$	$2$
60	$276$	$2^2\cdot 3\cdot 23$	2	$2^4{3}^4{23}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$		$10$
61	$281$		1b	$5^2{281}^4$	$1$	$3$	$5\ast$	$9\ast$	$2$	$(-,-,\otimes ,-)$	${\alpha }_1$	${𝔎}_1,{𝔎}_2$	$1$
62	$285$	$3\cdot 5\cdot 19$	1a	$5^6{3}^4{19}^4$	$16$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$		$1$
63	$286$	$2\cdot 11\cdot 13$	1b	$5^2{2}^4{11}^4{13}^4$	$13$	$2$	$3$	$7$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	${11}^2\cdot 5,𝒦$	$3$
64	$287$	$7\cdot 41$	1b	$5^2{7}^4{41}^4$	$4$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$	$1$
65	$290$	$2\cdot 5\cdot 29$	1a	$5^6{2}^4{29}^4$	$16$	$1$	$2$	$4$	$5$	$(-,\times ,-,-)$	$\epsilon$	$29$	$2$
66	$298$	$2\cdot 149$	1b	$5^2{2}^4{149}^4$	$4$	$1$	$2$	$4$	$5$	$(-,\times ,-,-)$	$\epsilon$	$5$	$2$
67	$301$	$7\cdot 43$	2	$7^4{43}^4$	$4$	$0$	$0$	$1$	$6$	$(-,-,-,\otimes )$	$\eta$		$1$
68	$302$	$2\cdot 151$	1b	$5^2{2}^4{151}^4$	$4$	$1$	$2$	$4$	$5$	$(-,-,\otimes ,-)$	${\beta }_1$	$2,{𝔎}_1$	$2$
69	$319$	$11\cdot 29$	1b	$5^2{11}^4{29}^4$	$3$	$2$	$2$	$5$	$2$	$(-,\otimes ,(\times ),-)$	${\alpha }_3$	${𝒦}_{\left(11\right)},{𝒦}_{\left(29\right)}$	$3$
70	$329$	$7\cdot 47$	1b	$5^2{7}^4{47}^4$	$4$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$47$	$7$
71	$330$	$2\cdot 3\cdot 5\cdot 11$	1a	$5^6{2}^4{3}^4{11}^4$	$64$	$2$	$4$	$9$	$6$	$(-,-,\times ,-)$	$\gamma$	$3\cdot 11,5\cdot {11}^3$	$2$
72	$341$	$11\cdot 31$	1b	$5^2{11}^4{31}^4$	$3$	$3$	$5$	$9$	$2$	$(-,-,\otimes ,-)$	${\alpha }_1$	${𝔎}_{\left(11\right),1}{𝔎}_{\left(31\right),2}^3,$
												${𝔎}_{\left(11\right),2}{𝔎}_{\left(31\right),1}$	$2$
73	$348$	$2^2\cdot 3\cdot 29$	1b	$5^2{2}^4{3}^4{29}^4$	$13$	$2$	$4$	$8$	$5$	$(-,\times ,-,-)$	$\epsilon$	$2\cdot 3\cdot 5$	$2$
74	$377$	$13\cdot 29$	1b	$5^2{13}^4{29}^4$	$3$	$2$	$3$	$6$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$𝒦$	$1$
75	$379$		1b	$5^2{379}^4$	$1$	$1$	$2$	$4$	$5$	$(-,\times ,-,-)$	$\epsilon$	$5$	$1$
76	$385$	$5\cdot 7\cdot 11$	1a	$5^6{7}^4{11}^4$	$16$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$7\cdot {11}^2,𝒦$	$1$
77	$390$	$2\cdot 3\cdot 5\cdot 13$	1a	$5^6{2}^4{3}^4{13}^4$	$64$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2,5$	$4$
78	$398$	$2\cdot 199$	1b	$5^2{2}^4{199}^4$	$4$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$5,𝒦$	$7$
79	$399$	$3\cdot 7\cdot 19$	2	$3^4{7}^4{19}^4$	$4$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$3\cdot {19}^2,𝒦$	$2$
80	$401$		2	${401}^4$	$1$	$2$	$3$	$5$	$2$	$(-,-,\otimes ,-)$	${\alpha }_1$	${𝔎}_1,{𝔎}_2$	$2$
81	$418$	$2\cdot 11\cdot 19$	2	$2^4{11}^4{19}^4$	$3$	$2$	$3$	$6$	$3$	$(-,\otimes ,\otimes ,-)$	${\alpha }_2$	${𝒦}_{\left(19\right)},{𝔎}_{\left(11\right),1}$	$1$
82	$421$		1b	$5^2{421}^4$	$1$	$1$	$2$	$3$	$4$	$(-,-,\otimes ,-)$	${\delta }_1$	${𝔎}_2$	$5$
83	$422$	$2\cdot 211$	1b	$5^2{2}^4{211}^4$	$3$	$1$	$2$	$4$	$5$	$(-,-,\times ,-)$	$\epsilon$	$2\cdot 5^2$	$6$
84	$451$	$11\cdot 41$	2	${11}^4{41}^4$	$1$	$2$	$3$	$6$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	${𝒦}_{\left(11\right)}{𝒦}_{\left(41\right)}^4,$
												${𝔎}_{\left(11\right),1}{𝔎}_{\left(41\right),2}^3$	$1$
85	$462$	$2\cdot 3\cdot 7\cdot 11$	1b	$5^2{2}^4{3}^4{7}^4{11}^4$	$52$	$2$	$4$	$9$	$6$	$(-,-,\times ,-)$	$\gamma$	$5^2\cdot 7,3\cdot 5\cdot {11}^2$	$1$
86	$465$	$3\cdot 5\cdot 31$	1a	$5^6{3}^4{31}^4$	$16$	$2$	$3$	$7\ast$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$5,𝒦$	$1$
87	$473$	$11\cdot 43$	1b	$5^2{11}^4{43}^4$	$4$	$2$	$3$	$7\ast$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$11,ℒ$	$1$
88	$482$	$2\cdot 241$	2	$2^4{241}^4$	$1$	$1$	$2$	$4$	$5$	$(-,-,\otimes ,-)$	${\beta }_1$	$241,{𝔎}_2$	$2$
89	$502$	$2\cdot 251$	1b	$5^2{2}^4{251}^4$	$4$	*$2$	*$4$	*$8$	$5$	$(-,-,\otimes ,-)$	${\beta }_1$	$251\cdot 5,{𝔎}_2$	$1$
90	$505$	$5\cdot 101$	1a	$5^6{101}^4$	$4$	$1$	$1$	$3$	$4$	$(-,-,(\times ),\otimes )$	${\zeta }_2$	$𝒦$	$2$

Table 29. Table 29. 39 39 39 prototypes 510 < 𝐌 < 900 510 𝐌 900 510<\mathbf{M}<900 of pure metacyclic fields

No.	$𝐌$	Factors	S	$f^4$	$m$	$V_L$	$V_M$	$V_N$	$E$	$(1,2,4,5)$	T	P	$\left|𝐌\right|$
91	$517$	$11\cdot 47$	1b	$5^2{11}^4{47}^4$	$3$	$2$	$3$	$6$	$3$	$(-,-,\otimes ,-)$	${\beta }_1$	$11\cdot 5^2,{𝔎}_2$	$1$
92	$532$	$2^2\cdot 7\cdot 19$	2	$2^4{7}^4{19}^4$	$4$	$1$	$2$	$4$	$5$	$(-,\times ,-,-)$	$\epsilon$	$2\cdot 7$	$1$
93	$546$	$2\cdot 3\cdot 7\cdot 13$	1b	$5^2{2}^4{3}^4{7}^4{13}^4$	$52$	$2$	$4$	$9$	$6$	$(\times ,-,-,-)$	$\gamma$	$3\cdot 5\cdot 13,$
												$5^2\cdot 7\cdot 13$	$3$
94	$550$	$2\cdot 5^2\cdot 11$	1a	$5^6{2}^4{11}^4$	$16$	$2$	$3$	$6$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$	$1$
95	$551$	$19\cdot 29$	2	${19}^4{29}^4$	$1$	$2$	$2$	$5$	$2$	$(-,\otimes ,-,-)$	${\alpha }_3$	${𝒦}_{\left(19\right)},{𝒦}_{\left(29\right)}$	$1$
96	$570$	$2\cdot 3\cdot 5\cdot 19$	1a	$5^6{2}^4{3}^4{19}^4$	$64$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$2^4\cdot 3,2^4\cdot 5$	$2$
97	$574$	$2\cdot 7\cdot 41$	2	$2^4{7}^4{41}^4$	$4$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$41,ℒ$	$3$
98	$590$	$2\cdot 5\cdot 59$	1a	$5^6{2}^4{59}^4$	$16$	$2$	$3$	*$7$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$5,𝒦$	$1$
99	$602$	$2\cdot 7\cdot 43$	1b	$5^2{2}^4{7}^4{43}^4$	$16$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2,7$	$2$
100	$606$	$2\cdot 3\cdot 101$	1b	$5^2{2}^4{3}^4{101}^4$	$12$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$2\cdot 5,101$	$2$
101	$609$	$3\cdot 7\cdot 29$	1b	$5^2{3}^4{7}^4{29}^4$	$12$	$2$	$3$	$7$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$7^2\cdot 5,𝒦$	$1$
102	$620$	$2^2\cdot 5\cdot 31$	1a	$5^6{2}^4{31}^4$	$16$	$2$	$4\ast$	$8\ast$	$5$	$(-,-,\otimes ,-)$	${\beta }_1$	$5,{𝔎}_1$	$1$
103	$627$	$3\cdot 11\cdot 19$	2	$5^2{3}^4{11}^4{59}^4$	$13$	$3$	$4$	$9$	$2$	$(-,\otimes ,(\times ),-)$	${\alpha }_3$	${𝒦}_{\left(11\right)},{𝒦}_{\left(19\right)}$	$1$
104	$638$	$2\cdot 11\cdot 29$	1b	$5^2{2}^4{11}^4{29}^4$	$13$	$2$	$3$	$7$	$4$	$(-,\otimes ,(\times ),-)$	${\beta }_2$	$2^4\cdot 11\cdot 5,$
												${𝒦}_{\left(11\right)}\cdot {𝒦}_{\left(29\right)}^4$	$2$
105	$649$	$11\cdot 59$	2	${11}^4{59}^4$	$1$	$2$	$2$	$5$	$2$	$(-,\otimes ,(\times ),-)$	${\alpha }_3$	${𝒦}_{\left(11\right)},{𝒦}_{\left(59\right)}$	$2$
106	$660$	$2^2\cdot 3\cdot 5\cdot 11$	1a	$5^6{2}^4{3}^4{11}^4$	$64$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$2\cdot 5,5\cdot 11$	$2$
107	$665$	$5\cdot 7\cdot 19$	1a	$5^6{7}^4{19}^4$	$16$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$7,𝒦$	$1$
108	$671$	$11\cdot 61$	1b	$5^2{11}^4{61}^4$	$3$	$3$	$4$	$8$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	${𝒦}_{\left(11\right)}{𝒦}_{\left(61\right)},$
												${𝔎}_{\left(11\right),2}{𝔎}_{\left(61\right),1}^2$	$1$
109	$682$	$2\cdot 11\cdot 31$	2	$2^4{11}^4{31}^4$	$3$	$2$	$3$	$6$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	${𝒦}_{\left(11\right)}{𝒦}_{\left(31\right)}^3,$
												${𝔎}_{\left(11\right),1}{𝔎}_{\left(31\right),2}^3$	$1$
110	$691$		1b	$5^2{691}^4$	$1$	$3$	$4$	$8$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$ℒ,{𝔎}_2$	$1$
111	$693$	$3^2\cdot 7\cdot 11$	2	$3^4{7}^4{11}^4$	$4$	$2$	$3$	$6$	$3$	$(-,-,\otimes ,-)$	${\beta }_1$	$3\cdot 7,{𝔎}_2$	$1$
112	$695$	$5\cdot 139$	1a	$5^6{139}^4$	$4$	$1$	$2$	$4$	$5$	$(-,\times ,-,-)$	$\epsilon$	$139$	$1$
113	$702$	$2\cdot 3^3\cdot 13$	1b	$5^2{2}^4{3}^4{13}^4$	$13$	$2$	$4$	$8$	$5$	$(\times ,-,-,-)$	$\epsilon$	$13\cdot 5^4$	$2$
114	$707$	$7\cdot 101$	2	$7^4{101}^4$	$4$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,\otimes )$	${\zeta }_2$	$𝒦$	$1$
115	$710$	$2\cdot 5\cdot 71$	1a	$5^6{2}^4{71}^4$	$16$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_2$	$4$
116	$745$	$5\cdot 149$	1a	$5^6{149}^4$	$4$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,\otimes )$	${\zeta }_2$	$𝒦$	$2$
117	$749$	$7\cdot 107$	2	$7^4{107}^4$	$4$	$1$	$2$	$4$	$5$	$(-,-,-,\times )$	$\epsilon$		$1$
118	$751$		2	${751}^4$	$1$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,\times )$	${\alpha }_2$	$ℒ,{𝔎}_1$	$1$
119	$770$	$2\cdot 5\cdot 7\cdot 11$	1a	$5^6{2}^4{7}^4{11}^4$	$64$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$2\cdot 5,2^2\cdot 11$	$1$
120	$779$	$19\cdot 41$	1b	$5^2{19}^4{41}^4$	$3$	$3$	$4$	$8$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	${𝒦}_{\left(19\right)}{𝒦}_{\left(41\right)},$
												${𝔎}_{\left(41\right),1}$	$1$
121	$785$	$5\cdot 157$	1a	$5^6{157}^4$	$4$	$1$	$2$	$4$	$5$	$(-,-,-,\times )$	$\epsilon$		$1$
122	$798$	$2\cdot 3\cdot 7\cdot 19$	1b	$5^2{2}^4{3}^4{7}^4{19}^4$	$52$	$2$	$4$	$9$	$6$	$(-,\times ,-,-)$	$\gamma$	$3\cdot 19,7\cdot 5^4$	$1$
123	$808$	$2^3\cdot 101$	1b	$5^2{2}^4{101}^4$	$4$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$	$2$
124	$825$	$3\cdot 5^2\cdot 11$	1a	$5^6{3}^4{11}^4$	$16$	$1$	$2$	$4$	$5$	$(-,-,\otimes ,-)$	${\beta }_1$	$3^2\cdot 11,{𝔎}_1$	$1$
125	$843$	$3\cdot 281$	2	$3^4{281}^4$	$1$	$1$	$2$	$3$	$4$	$(-,-,\otimes ,-)$	${\delta }_1$	${𝔎}_1$	$1$
126	$858$	$2\cdot 3\cdot 11\cdot 13$	1b	$5^2{2}^4{3}^4{11}^4{13}^4$	$51$	$2$	$4$	$9$	$6$	$(-,-,\times ,-)$	$\gamma$	$2\cdot 5,13\cdot 5$	$1$
127	$861$	$3\cdot 7\cdot 41$	1b	$5^2{3}^4{7}^4{41}^4$	$12$	$3$	$4$	$8$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$	$1$
128	$893$	$19\cdot 47$	2	${19}^4{47}^4$	$1$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$19,𝔏$	$1$
129	$894$	$2\cdot 3\cdot 149$	1b	$5^2{2}^4{3}^4{149}^4$	$12$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$5,2\cdot 3^2$	$1$

Table 30. Table 30. 5 5 5 prototypes 900 < 𝐌 < 1000 900 𝐌 1000 900<\mathbf{M}<1000 of pure metacyclic fields

No.	$𝐌$	Factors	S	$f^4$	$m$	$V_L$	$V_M$	$V_N$	$E$	$(1,2,4,5)$	T	P	$\left|𝐌\right|$
130	$902$	$2\cdot 11\cdot 41$	1b	$5^2{2}^4{11}^4{41}^4$	$13$	$2$	$3$	$7$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$11\cdot 5,$
												${𝒦}_{\left(11\right)}\cdot {𝒦}_{\left(41\right)}$	$1$
131	$924$	$2^2\cdot 3\cdot 7\cdot 11$	1a	$2^4{3}^4{7}^4{11}^4$	$12$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$3\cdot 7,7\cdot 11$	$1$
132	$955$	$5\cdot 191$	1a	$5^6{191}^4$	$4$	$2\ast$	$3\ast$	$6\ast$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$	$1$
133	$957$	$3\cdot 11\cdot 29$	2	$3^4{11}^4{29}^4$	$3$	$2$	$2$	$5$	$2$	$(-,\otimes ,(\times ),-)$	${\alpha }_3$	${𝒦}_{\left(11\right)},{𝒦}_{\left(29\right)}$	$1$
134	$982$	$2\cdot 491$	2	$2^4{491}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$	$2$

Table 31. Table 31. Splitting of type α 1 subscript 𝛼 1 \alpha_{1} , ( U , η , ζ ; A , I , R ) = ( 2 , − , − ; 1 , 0 , 2 ) 𝑈 𝜂 𝜁 𝐴 𝐼 𝑅 2 1 0 2 (U,\eta,\zeta;A,I,R)=(2,-,-;1,0,2) : 5 5 5 similarity classes

No.	S	$e_0$	$t$	$u$	$v$	$m$	$n$	$s_2$	$s_4$	$V_L$	$V_M$	$V_N$	$E$	$𝐌$	$\left|𝐌\right|$
1	1b	$2$	$1$	$0$	$1$	$1$	$0$	$0$	$1$	$2$	$3$	$5$	$2$	$\mathrm{𝟑𝟏}$	$2$
2	1a	$6$	$1$	$0$	$1$	$4$	$0$	$0$	$1$	$2$	$3$	$5$	$2$	$\mathrm{𝟏𝟓𝟓}$	$2$
3	1b	$2$	$1$	$0$	$1$	$1$	$0$	$0$	$1$	$3$	$5\ast$	$9\ast$	$2$	$\mathrm{𝟐𝟖𝟏}$	$1$
4	1b	$2$	$2$	$0$	$2$	$3$	$0$	$0$	$2$	$3$	$5$	$9$	$2$	$\mathrm{𝟑𝟒𝟏}$	$2$
5	2	$0$	$1$	$1^{\prime }$	$0$	$1$	$0$	$0$	$1^{\prime }$	$2$	$3$	$5$	$2$	$\mathrm{𝟒𝟎𝟏}$	$2$

Table 32. Table 32. Splitting of type α 2 subscript 𝛼 2 \alpha_{2} , ( U , η , ζ ; A , I , R ) = ( 2 , − , − ; 1 , 1 , 1 ) 𝑈 𝜂 𝜁 𝐴 𝐼 𝑅 2 1 1 1 (U,\eta,\zeta;A,I,R)=(2,-,-;1,1,1) : 22 22 22 similarity classes

No.	S	$e_0$	$t$	$u$	$v$	$m$	$n$	$s_2$	$s_4$	$V_L$	$V_M$	$V_N$	$E$	$𝐌$	$\left|𝐌\right|$
1	1b	$2$	$1$	$0$	$1$	$1$	$0$	$0$	$1$	$1$	$1$	$2$	$3$	$\mathrm{𝟏𝟏}$	$14$
2	1b	$2$	$2$	$0$	$2$	$3$	$1$	$0$	$1$	$2$	$2$	$4$	$1$	$\mathrm{𝟑𝟑}$	$8$
3	1a	$6$	$1$	$0$	$1$	$4$	$0$	$0$	$1$	$1$	$1$	$2$	$3$	$\mathrm{𝟓𝟓}$	$6$
4	1b	$2$	$2$	$0$	$2$	$3$	$1$	$0$	$1$	$1$	$1$	$2$	$3$	$\mathrm{𝟖𝟐}$	$15$
5	1b	$2$	$2$	$0$	$2$	$3$	$1$	$0$	$1$	$2$	$3$	$6$	$3$	$\mathrm{𝟏𝟐𝟑}$	$2$
6	1b	$2$	$1$	$0$	$1$	$1$	$0$	$0$	$1$	$2$	$2$	$4$	$1$	$\mathrm{𝟏𝟑𝟏}$	$6$
7	2	$0$	$1$	$1^{\prime }$	$0$	$1$	$0$	$0$	$1^{\prime }$	$1$	$1$	$2$	$3$	$\mathrm{𝟏𝟓𝟏}$	$3$
8	1a	$6$	$1$	$0$	$1$	$4$	$0$	$0$	$1$	$2$	$2$	$4$	$1$	$\mathrm{𝟐𝟕𝟓}$	$2$
9	1b	$2$	$2$	$1$	$1$	$4$	$1$	$0$	$1$	$2$	$2$	$4$	$1$	$\mathrm{𝟐𝟖𝟕}$	$1$
10	2	$0$	$3$	$0$	$3$	$3$	$1$	$1$	$1$	$2$	$3$	$6$	$3$	$\mathrm{𝟒𝟏𝟖}$	$1$
11	2	$0$	$2$	$0$	$2$	$1$	$0$	$0$	$2$	$2$	$3$	$6$	$3$	$\mathrm{𝟒𝟓𝟏}$	$1$
12	1a	$6$	$2$	$0$	$2$	$16$	$1$	$0$	$1$	$2$	$3$	$6$	$3$	$\mathrm{𝟓𝟓𝟎}$	$1$
13	1b	$2$	$2$	$0$	$2$	$3$	$0$	$0$	$2$	$3$	$4$	$8$	$1$	$\mathrm{𝟔𝟕𝟏}$	$1$
14	2	$0$	$3$	$0$	$3$	$3$	$1$	$0$	$2$	$2$	$3$	$6$	$3$	$\mathrm{𝟔𝟖𝟐}$	$1$
15	1b	$2$	$1$	$0$	$1$	$1$	$0$	$0$	$1$	$3$	$4$	$8$	$1$	$\mathrm{𝟔𝟗𝟏}$	$1$
16	1a	$6$	$2$	$0$	$2$	$16$	$1$	$0$	$1$	$2$	$2$	$4$	$1$	$\mathrm{𝟕𝟏𝟎}$	$4$
17	2	$0$	$1$	$1^{\prime }$	$0$	$1$	$0$	$0$	$1^{\prime }$	$2$	$2$	$4$	$1$	$\mathrm{𝟕𝟓𝟏}$	$1$
18	1b	$2$	$2$	$0$	$2$	$3$	$0$	$1$	$1$	$3$	$4$	$8$	$1$	$\mathrm{𝟕𝟕𝟗}$	$1$
19	1b	$2$	$2$	$1^{\prime }$	$1$	$4$	$1$	$0$	$1^{\prime }$	$2$	$2$	$4$	$1$	$\mathrm{𝟖𝟎𝟖}$	$2$
20	1b	$2$	$3$	$1$	$2$	$12$	$2$	$0$	$1$	$3$	$4$	$8$	$1$	$\mathrm{𝟖𝟔𝟏}$	$1$
21	1a	$6$	$1$	$0$	$1$	$4$	$0$	$0$	$1$	$2\ast$	$3\ast$	$6\ast$	$3$	$\mathrm{𝟗𝟓𝟓}$	$1$
22	2	$0$	$2$	$0$	$2$	$1$	$1$	$0$	$1$	$1$	$1$	$2$	$3$	$\mathrm{𝟗𝟖𝟐}$	$2$

Table 33. Table 33. Splitting of type α 3 subscript 𝛼 3 \alpha_{3} , ( U , η , ζ ; A , I , R ) = ( 2 , − , − ; 1 , 2 , 0 ) 𝑈 𝜂 𝜁 𝐴 𝐼 𝑅 2 1 2 0 (U,\eta,\zeta;A,I,R)=(2,-,-;1,2,0) : 5 5 5 similarity classes

No.	S	$e_0$	$t$	$u$	$v$	$m$	$n$	$s_2$	$s_4$	$V_L$	$V_M$	$V_N$	$E$	$𝐌$	$\left|𝐌\right|$
1	1b	$2$	$2$	$0$	$2$	$3$	$0$	$1$	$1$	$2$	$2$	$5$	$2$	$\mathrm{𝟑𝟏𝟗}$	$3$
2	2	$0$	$2$	$0$	$2$	$1$	$0$	$2$	$0$	$2$	$2$	$5$	$2$	$\mathrm{𝟓𝟓𝟏}$	$1$
3	1b	$2$	$3$	$0$	$3$	$13$	$1$	$1$	$1$	$3$	$4$	$9$	$2$	$\mathrm{𝟔𝟐𝟕}$	$1$
4	2	$0$	$2$	$0$	$2$	$1$	$0$	$1$	$1$	$2$	$2$	$5$	$2$	$\mathrm{𝟔𝟒𝟗}$	$2$
5	2	$0$	$3$	$0$	$3$	$3$	$1$	$1$	$1$	$2$	$2$	$5$	$2$	$\mathrm{𝟗𝟓𝟕}$	$1$

Table 34. Table 34. Splitting of type β 1 subscript 𝛽 1 \beta_{1} , ( U , η , ζ ; A , I , R ) = ( 2 , − , − ; 2 , 0 , 1 ) 𝑈 𝜂 𝜁 𝐴 𝐼 𝑅 2 2 0 1 (U,\eta,\zeta;A,I,R)=(2,-,-;2,0,1) : 10 10 10 similarity classes

No.	S	$e_0$	$t$	$u$	$v$	$m$	$n$	$s_2$	$s_4$	$V_L$	$V_M$	$V_N$	$E$	$𝐌$	$\left|𝐌\right|$
1	1b	$2$	$3$	$0$	$3$	$13$	$2$	$0$	$1$	$2$	$3$	$6$	$3$	$\mathrm{𝟏𝟖𝟔}$	$7$
2	1b	$2$	$1$	$0$	$1$	$1$	$0$	$0$	$1$	$1$	$2$	$4$	$5$	$\mathrm{𝟏𝟗𝟏}$	$3$
3	1b	$2$	$2$	$0$	$2$	$3$	$1$	$0$	$1$	$1$	$2$	$4$	$5$	$\mathrm{𝟐𝟓𝟑}$	$4$
4	1b	$2$	$2$	$1^{\prime }$	$1$	$4$	$1$	$0$	$1^{\prime }$	$1$	$2$	$4$	$5$	$\mathrm{𝟑𝟎𝟐}$	$2$
5	2	$0$	$2$	$0$	$2$	$1$	$1$	$0$	$1$	$1$	$2$	$4$	$5$	$\mathrm{𝟒𝟖𝟐}$	$2$
6	1b	$2$	$2$	$1^{\prime }$	$1$	$4$	$1$	$0$	$1^{\prime }$	$2\ast$	$4\ast$	$8\ast$	$5$	$\mathrm{𝟓𝟎𝟐}$	$1$
7	1b	$2$	$2$	$0$	$2$	$3$	$1$	$0$	$1$	$2$	$3$	$6$	$3$	$\mathrm{𝟓𝟏𝟕}$	$1$
8	1a	$6$	$2$	$0$	$2$	$16$	$1$	$0$	$1$	$2$	$4\ast$	$8\ast$	$5$	$\mathrm{𝟔𝟐𝟎}$	$1$
9	2	$0$	$3$	$1$	$2$	$4$	$2$	$0$	$1$	$2$	$3$	$6$	$3$	$\mathrm{𝟔𝟗𝟑}$	$1$
10	1a	$6$	$2$	$0$	$2$	$16$	$1$	$0$	$1$	$1$	$2$	$4$	$5$	$\mathrm{𝟖𝟐𝟓}$	$1$

Table 35. Table 35. Splitting of type β 2 subscript 𝛽 2 \beta_{2} , ( U , η , ζ ; A , I , R ) = ( 2 , − , − ; 2 , 1 , 0 ) 𝑈 𝜂 𝜁 𝐴 𝐼 𝑅 2 2 1 0 (U,\eta,\zeta;A,I,R)=(2,-,-;2,1,0) : 25 25 25 similarity classes

No.	S	$e_0$	$t$	$u$	$v$	$m$	$n$	$s_2$	$s_4$	$V_L$	$V_M$	$V_N$	$E$	$𝐌$	$\left|𝐌\right|$
1	1b	$2$	$2$	$0$	$2$	$3$	$1$	$0$	$1$	$1$	$1$	$3$	$4$	$\mathrm{𝟐𝟐}$	$35$
2	1b	$2$	$2$	$0$	$2$	$3$	$1$	$1$	$0$	$1$	$1$	$3$	$4$	$\mathrm{𝟑𝟖}$	$44$
3	1b	$2$	$2$	$1$	$1$	$4$	$1$	$0$	$1$	$1$	$1$	$3$	$4$	$\mathrm{𝟕𝟕}$	$7$
4	1a	$6$	$2$	$0$	$2$	$16$	$1$	$0$	$1$	$1$	$1$	$3$	$4$	$\mathrm{𝟏𝟏𝟎}$	$11$
5	2	$0$	$3$	$0$	$3$	$3$	$2$	$0$	$1$	$1$	$1$	$3$	$4$	$\mathrm{𝟏𝟑𝟐}$	$5$
6	1b	$2$	$2$	$1$	$1$	$4$	$1$	$1$	$0$	$1$	$1$	$3$	$4$	$\mathrm{𝟏𝟑𝟑}$	$7$
7	1b	$2$	$1$	$0$	$1$	$1$	$0$	$1$	$0$	$1$	$1$	$3$	$4$	$\mathrm{𝟏𝟑𝟗}$	$4$
8	1b	$2$	$3$	$1$	$2$	$12$	$2$	$0$	$1$	$2$	$3$	$7$	$4$	$\mathrm{𝟏𝟓𝟒}$	$3$
9	2	$0$	$3$	$0$	$3$	$3$	$2$	$1$	$0$	$1$	$1$	$3$	$4$	$\mathrm{𝟏𝟕𝟒}$	$3$
10	1a	$6$	$2$	$0$	$2$	$16$	$1$	$1$	$0$	$1$	$1$	$3$	$4$	$\mathrm{𝟏𝟗𝟎}$	$9$
11	1b	$2$	$2$	$1^{\prime }$	$1$	$4$	$1$	$0$	$1^{\prime }$	$1$	$1$	$3$	$4$	$\mathrm{𝟐𝟎𝟐}$	$6$
12	1b	$2$	$2$	$0$	$2$	$3$	$0$	$1$	$1$	$2$	$3$	$7$	$4$	$\mathrm{𝟐𝟎𝟗}$	$2$
13	1b	$2$	$3$	$0$	$3$	$13$	$2$	$0$	$1$	$2$	$3$	$7$	$4$	$\mathrm{𝟐𝟖𝟔}$	$3$
14	1a	$6$	$2$	$1$	$1$	$16$	$1$	$0$	$1$	$1$	$1$	$3$	$4$	$\mathrm{𝟑𝟖𝟓}$	$1$
15	1b	$2$	$2$	$1^{\prime }$	$1$	$4$	$1$	$1^{\prime }$	$0$	$1$	$1$	$3$	$4$	$\mathrm{𝟑𝟗𝟖}$	$7$
16	2	$0$	$3$	$1$	$2$	$4$	$2$	$1$	$0$	$1$	$1$	$3$	$4$	$\mathrm{𝟑𝟗𝟗}$	$2$
17	1a	$6$	$2$	$0$	$2$	$16$	$1$	$0$	$1$	$2$	$3$	$7\ast$	$4$	$\mathrm{𝟒𝟔𝟓}$	$1$
18	1b	$2$	$2$	$1$	$1$	$4$	$1$	$0$	$1$	$2$	$3$	$7\ast$	$4$	$\mathrm{𝟒𝟕𝟑}$	$1$
19	2	$0$	$3$	$1$	$2$	$4$	$2$	$0$	$1$	$1$	$1$	$3$	$4$	$\mathrm{𝟓𝟕𝟒}$	$3$
20	1a	$6$	$2$	$0$	$2$	$16$	$1$	$1$	$0$	$2$	$3$	$7\ast$	$4$	$\mathrm{𝟓𝟗𝟎}$	$1$
21	1b	$2$	$3$	$1$	$2$	$12$	$2$	$1$	$0$	$2$	$3$	$7$	$4$	$\mathrm{𝟔𝟎𝟗}$	$1$
22	1b	$2$	$3$	$0$	$3$	$13$	$1$	$1$	$1$	$2$	$3$	$7$	$4$	$\mathrm{𝟔𝟑𝟖}$	$2$
23	1a	$6$	$2$	$1$	$1$	$16$	$1$	$1$	$0$	$1$	$1$	$3$	$4$	$\mathrm{𝟔𝟔𝟓}$	$1$
24	2	$0$	$2$	$0$	$2$	$1$	$1$	$1$	$0$	$1$	$1$	$3$	$4$	$\mathrm{𝟖𝟗𝟑}$	$1$
25	1b	$2$	$3$	$0$	$3$	$13$	$1$	$0$	$2$	$2$	$3$	$7$	$4$	$\mathrm{𝟗𝟎𝟐}$	$1$

Table 36. Table 36. Splitting of type γ 𝛾 \gamma , ( U , η , ζ ; A , I , R ) = ( 2 , − , − ; 3 , 0 , 0 ) 𝑈 𝜂 𝜁 𝐴 𝐼 𝑅 2 3 0 0 (U,\eta,\zeta;A,I,R)=(2,-,-;3,0,0) : 29 29 29 similarity classes

No.	S	$e_0$	$t$	$u$	$v$	$m$	$n$	$s_2$	$s_4$	$V_L$	$V_M$	$V_N$	$E$	$𝐌$	$\left|𝐌\right|$
1	1b	$2$	$2$	$0$	$2$	$3$	$2$	$0$	$0$	$0$	$0$	$1$	$6$	$\mathrm{𝟔}$	$77$
2	1b	$2$	$2$	$1$	$1$	$4$	$2$	$0$	$0$	$0$	$0$	$1$	$6$	$\mathrm{𝟏𝟒}$	$44$
3	1a	$6$	$2$	$0$	$2$	$16$	$2$	$0$	$0$	$0$	$0$	$1$	$6$	$\mathrm{𝟑𝟎}$	$37$
4	1b	$2$	$3$	$1$	$2$	$12$	$3$	$0$	$0$	$1$	$2$	$5$	$6$	$\mathrm{𝟒𝟐}$	$22$
5	1b	$2$	$3$	$0$	$3$	$13$	$2$	$0$	$1$	$1$	$2$	$5$	$6$	$\mathrm{𝟔𝟔}$	$17$
6	1a	$6$	$2$	$1$	$1$	$16$	$2$	$0$	$0$	$0$	$0$	$1$	$6$	$\mathrm{𝟕𝟎}$	$14$
7	1b	$2$	$3$	$0$	$3$	$13$	$3$	$0$	$0$	$1$	$2$	$5$	$6$	$\mathrm{𝟕𝟖}$	$37$
8	1b	$2$	$3$	$0$	$3$	$13$	$2$	$1$	$0$	$1$	$2$	$5$	$6$	$\mathrm{𝟏𝟏𝟒}$	$20$
9	2	$0$	$3$	$1$	$2$	$4$	$3$	$0$	$0$	$0$	$0$	$1$	$6$	$\mathrm{𝟏𝟐𝟔}$	$6$
10	1a	$6$	$3$	$1$	$2$	$64$	$3$	$0$	$0$	$1$	$2$	$5$	$6$	$\mathrm{𝟐𝟏𝟎}$	$5$
11	1b	$2$	$3$	$1$	$2$	$12$	$2$	$0$	$1$	$1$	$2$	$5$	$6$	$\mathrm{𝟐𝟑𝟏}$	$5$
12	1b	$2$	$2$	$0$	$2$	$3$	$1$	$1$	$0$	$1$	$2$	$5$	$6$	$\mathrm{𝟐𝟒𝟕}$	$2$
13	1b	$2$	$2$	$1$	$1$	$4$	$2$	$0$	$0$	$1$	$2$	$5$	$6$	$\mathrm{𝟐𝟓𝟗}$	$1$
14	1b	$2$	$3$	$1$	$2$	$12$	$2$	$1$	$0$	$1$	$2$	$5$	$6$	$\mathrm{𝟐𝟔𝟔}$	$3$
15	2	$0$	$3$	$0$	$3$	$3$	$3$	$0$	$0$	$0$	$0$	$1$	$6$	$\mathrm{𝟐𝟕𝟔}$	$10$
16	1a	$6$	$2$	$0$	$2$	$16$	$1$	$1$	$0$	$1$	$2$	$5$	$6$	$\mathrm{𝟐𝟖𝟓}$	$1$
17	1a	$6$	$3$	$0$	$3$	$64$	$2$	$0$	$1$	$2$	$4$	$9$	$6$	$\mathrm{𝟑𝟑𝟎}$	$2$
18	1a	$6$	$3$	$0$	$3$	$64$	$3$	$0$	$0$	$1$	$2$	$5$	$6$	$\mathrm{𝟑𝟗𝟎}$	$4$
19	1b	$2$	$4$	$1$	$3$	$52$	$3$	$0$	$1$	$2$	$4$	$9$	$6$	$\mathrm{𝟒𝟔𝟐}$	$1$
20	1b	$2$	$4$	$1$	$3$	$52$	$4$	$0$	$0$	$2$	$4$	$9$	$6$	$\mathrm{𝟓𝟒𝟔}$	$3$
21	1a	$6$	$3$	$0$	$3$	$64$	$2$	$1$	$0$	$1$	$2$	$5$	$6$	$\mathrm{𝟓𝟕𝟎}$	$2$
22	1b	$2$	$3$	$2$	$1$	$16$	$3$	$0$	$0$	$1$	$2$	$5$	$6$	$\mathrm{𝟔𝟎𝟐}$	$2$
23	1b	$2$	$3$	$1^{\prime }$	$2$	$12$	$2$	$0$	$1^{\prime }$	$1$	$2$	$5$	$6$	$\mathrm{𝟔𝟎𝟔}$	$2$
24	1a	$6$	$3$	$0$	$3$	$64$	$2$	$0$	$1$	$1$	$2$	$5$	$6$	$\mathrm{𝟔𝟔𝟎}$	$2$
25	1a	$6$	$3$	$1$	$2$	$64$	$2$	$0$	$1$	$1$	$2$	$5$	$6$	$\mathrm{𝟕𝟕𝟎}$	$1$
26	1b	$2$	$4$	$1$	$3$	$52$	$3$	$1$	$0$	$2$	$4$	$9$	$6$	$\mathrm{𝟕𝟗𝟖}$	$1$
27	1b	$2$	$4$	$0$	$4$	$51$	$3$	$0$	$1$	$2$	$4$	$9$	$6$	$\mathrm{𝟖𝟓𝟖}$	$1$
28	1b	$2$	$3$	$1^{\prime }$	$2$	$12$	$2$	$1^{\prime }$	$0$	$1$	$2$	$5$	$6$	$\mathrm{𝟖𝟗𝟒}$	$1$
29	2	$0$	$4$	$1$	$3$	$12$	$3$	$0$	$1$	$1$	$2$	$5$	$6$	$\mathrm{𝟗𝟐𝟒}$	$1$

Table 37. Table 37. Splitting of type δ 1 subscript 𝛿 1 \delta_{1} , ( U , η , ζ ; A , I , R ) = ( 1 , × , − ; 1 , 0 , 1 ) 𝑈 𝜂 𝜁 𝐴 𝐼 𝑅 1 1 0 1 (U,\eta,\zeta;A,I,R)=(1,\times,-;1,0,1) : 3 3 3 similarity classes

No.	S	$e_0$	$t$	$u$	$v$	$m$	$n$	$s_2$	$s_4$	$V_L$	$V_M$	$V_N$	$E$	$𝐌$	$\left|𝐌\right|$
1	1b	$2$	$1$	$0$	$1$	$1$	$0$	$0$	$1$	$3$	$5$	$9$	$2$	$\mathrm{𝟐𝟏𝟏}$	$1$
2	1b	$2$	$1$	$0$	$1$	$1$	$0$	$0$	$1$	$1$	$2$	$3$	$4$	$\mathrm{𝟒𝟐𝟏}$	$5$
3	2	$0$	$2$	$0$	$2$	$1$	$1$	$0$	$1$	$1$	$2$	$3$	$4$	$\mathrm{𝟖𝟒𝟑}$	$1$

Table 38. Table 38. Splitting of type δ 2 subscript 𝛿 2 \delta_{2} , ( U , η , ζ ; A , I , R ) = ( 1 , × , − ; 1 , 1 , 0 ) 𝑈 𝜂 𝜁 𝐴 𝐼 𝑅 1 1 1 0 (U,\eta,\zeta;A,I,R)=(1,\times,-;1,1,0) : 5 5 5 similarity classes

No.	S	$e_0$	$t$	$u$	$v$	$m$	$n$	$s_2$	$s_4$	$V_L$	$V_M$	$V_N$	$E$	$𝐌$	$\left|𝐌\right|$
1	1b	$2$	$1$	$0$	$1$	$1$	$0$	$1$	$0$	$1$	$1$	$2$	$3$	$\mathrm{𝟏𝟗}$	$27$
2	2	$0$	$2$	$0$	$2$	$1$	$1$	$1$	$0$	$1$	$1$	$2$	$3$	$\mathrm{𝟓𝟕}$	$10$
3	1a	$6$	$1$	$0$	$1$	$4$	$0$	$1$	$0$	$1$	$1$	$2$	$3$	$\mathrm{𝟗𝟓}$	$9$
4	2	$0$	$1$	$1^{\prime }$	$0$	$1$	$0$	$1^{\prime }$	$0$	$1$	$1$	$2$	$3$	$\mathrm{𝟏𝟒𝟗}$	$6$
5	1b	$2$	$2$	$0$	$2$	$3$	$1$	$1$	$0$	$2$	$3$	$6$	$3$	$\mathrm{𝟑𝟕𝟕}$	$1$

Table 39. Table 39. Splitting of type ε 𝜀 \varepsilon , ( U , η , ζ ; A , I , R ) = ( 1 , × , − ; 2 , 0 , 0 ) 𝑈 𝜂 𝜁 𝐴 𝐼 𝑅 1 2 0 0 (U,\eta,\zeta;A,I,R)=(1,\times,-;2,0,0) : 22 22 22 similarity classes

No.	S	$e_0$	$t$	$u$	$v$	$m$	$n$	$s_2$	$s_4$	$V_L$	$V_M$	$V_N$	$E$	$𝐌$	$\left|𝐌\right|$
1	1b	$2$	$1$	$0$	$1$	$1$	$1$	$0$	$0$	$0$	$0$	$0$	$5$	$\mathrm{𝟐}$	$71$
2	1a	$6$	$1$	$0$	$1$	$4$	$1$	$0$	$0$	$0$	$0$	$0$	$5$	$\mathrm{𝟏𝟎}$	$31$
3	2	$0$	$2$	$0$	$2$	$1$	$2$	$0$	$0$	$0$	$0$	$0$	$5$	$\mathrm{𝟏𝟖}$	$37$
4	1a	$6$	$2$	$1$	$1$	$16$	$2$	$0$	$0$	$1$	$2$	$4$	$5$	$\mathrm{𝟏𝟒𝟎}$	$5$
5	1b	$2$	$2$	$0$	$2$	$3$	$2$	$0$	$0$	$1$	$2$	$4$	$5$	$\mathrm{𝟏𝟒𝟏}$	$19$
6	1b	$2$	$2$	$0$	$2$	$3$	$1$	$1$	$0$	$1$	$2$	$4$	$5$	$\mathrm{𝟏𝟕𝟏}$	$6$
7	1a	$6$	$2$	$0$	$2$	$16$	$2$	$0$	$0$	$1$	$2$	$4$	$5$	$\mathrm{𝟏𝟖𝟎}$	$5$
8	2	$0$	$3$	$1$	$2$	$4$	$2$	$0$	$0$	$1$	$2$	$4$	$5$	$\mathrm{𝟏𝟖𝟐}$	$1$
9	1b	$2$	$2$	$1$	$1$	$4$	$1$	$1$	$0$	$1$	$2$	$4$	$5$	$\mathrm{𝟐𝟎𝟑}$	$2$
10	2	$0$	$2$	$0$	$2$	$1$	$1$	$1$	$0$	$1$	$2$	$4$	$5$	$\mathrm{𝟐𝟏𝟖}$	$1$
11	1b	$2$	$3$	$1$	$2$	$12$	$3$	$0$	$0$	$2$	$4$	$8$	$5$	$\mathrm{𝟐𝟕𝟑}$	$4$
12	1a	$6$	$2$	$0$	$2$	$16$	$1$	$1$	$0$	$1$	$2$	$4$	$5$	$\mathrm{𝟐𝟗𝟎}$	$2$
13	1b	$2$	$2$	$0$	$2$	$3$	$1$	$1$	$0$	$1$	$2$	$4$	$5$	$\mathrm{𝟐𝟗𝟖}$	$2$
14	1b	$2$	$2$	$1$	$1$	$4$	$2$	$0$	$0$	$1$	$2$	$4$	$5$	$\mathrm{𝟑𝟐𝟗}$	$7$
15	1b	$2$	$3$	$0$	$3$	$13$	$2$	$1$	$0$	$2$	$4$	$8$	$5$	$\mathrm{𝟑𝟒𝟖}$	$2$
16	1b	$2$	$1$	$0$	$1$	$1$	$0$	$1$	$0$	$1$	$2$	$4$	$5$	$\mathrm{𝟑𝟕𝟗}$	$1$
17	1b	$2$	$2$	$0$	$2$	$3$	$1$	$0$	$1$	$1$	$2$	$4$	$5$	$\mathrm{𝟒𝟐𝟐}$	$6$
18	2	$0$	$3$	$1$	$2$	$4$	$2$	$1$	$0$	$1$	$2$	$4$	$5$	$\mathrm{𝟓𝟑𝟐}$	$1$
19	1a	$6$	$1$	$0$	$1$	$4$	$0$	$1$	$0$	$1$	$2$	$4$	$5$	$\mathrm{𝟔𝟗𝟓}$	$1$
20	1b	$2$	$3$	$0$	$3$	$13$	$3$	$0$	$0$	$2$	$4$	$8$	$5$	$\mathrm{𝟕𝟎𝟐}$	$2$
21	2	$0$	$2$	$2$	$0$	$4$	$2$	$0$	$0$	$1$	$2$	$4$	$5$	$\mathrm{𝟕𝟒𝟗}$	$1$
22	1a	$6$	$1$	$1$	$0$	$4$	$1$	$0$	$0$	$1$	$2$	$4$	$5$	$\mathrm{𝟕𝟖𝟓}$	$1$

Table 40. Table 40. No splitting of type ζ 1 subscript 𝜁 1 \zeta_{1} , ( U , η , ζ ; A , I , R ) = ( 1 , − , × ; 1 , 0 , 1 ) 𝑈 𝜂 𝜁 𝐴 𝐼 𝑅 1 1 0 1 (U,\eta,\zeta;A,I,R)=(1,-,\times;1,0,1) : 1 1 1 similarity class

No.	S	$e_0$	$t$	$u$	$v$	$m$	$n$	$s_2$	$s_4$	$V_L$	$V_M$	$V_N$	$E$	$𝐌$	$\left|𝐌\right|$
1	2	$0$	$1$	$1^{\prime }$	$0$	$1$	$0$	$0$	$1^{\prime }$	$1$	$2$	$4$	$5$	$\mathrm{𝟏𝟎𝟏}$	$1$

Table 41. Table 41. Splitting of type ζ 2 subscript 𝜁 2 \zeta_{2} , ( U , η , ζ ; A , I , R ) = ( 1 , − , × ; 1 , 1 , 0 ) 𝑈 𝜂 𝜁 𝐴 𝐼 𝑅 1 1 1 0 (U,\eta,\zeta;A,I,R)=(1,-,\times;1,1,0) : 3 3 3 similarity classes

No.	S	$e_0$	$t$	$u$	$v$	$m$	$n$	$s_2$	$s_4$	$V_L$	$V_M$	$V_N$	$E$	$𝐌$	$\left|𝐌\right|$
1	1a	$6$	$1$	$1^{\prime }$	$0$	$4$	$0$	$0$	$1^{\prime }$	$1$	$1$	$3$	$4$	$\mathrm{𝟓𝟎𝟓}$	$2$
2	2	$0$	$2$	$2^{\prime }$	$0$	$4$	$1$	$0$	$1^{\prime }$	$1$	$1$	$3$	$4$	$\mathrm{𝟕𝟎𝟕}$	$1$
3	1a	$6$	$1$	$1$	$0$	$4$	$0$	$1$	$0$	$1$	$1$	$3$	$4$	$\mathrm{𝟕𝟒𝟓}$	$2$

Table 42. Table 42. Splitting of type η 𝜂 \eta , ( U , η , ζ ; A , I , R ) = ( 1 , − , × ; 2 , 0 , 0 ) 𝑈 𝜂 𝜁 𝐴 𝐼 𝑅 1 2 0 0 (U,\eta,\zeta;A,I,R)=(1,-,\times;2,0,0) : 2 2 2 similarity classes

No.	S	$e_0$	$t$	$u$	$v$	$m$	$n$	$s_2$	$s_4$	$V_L$	$V_M$	$V_N$	$E$	$𝐌$	$\left|𝐌\right|$
1	1a	$6$	$1$	$1$	$0$	$4$	$1$	$0$	$0$	$0$	$0$	$1$	$6$	$\mathrm{𝟑𝟓}$	$6$
2	2	$0$	$2$	$2$	$0$	$4$	$2$	$0$	$0$	$0$	$0$	$1$	$6$	$\mathrm{𝟑𝟎𝟏}$	$1$

Table 43. Table 43. Splitting of type ϑ italic-ϑ \vartheta , ( U , η , ζ ; A , I , R ) = ( 0 , × , × ; 1 , 0 , 0 ) 𝑈 𝜂 𝜁 𝐴 𝐼 𝑅 0 1 0 0 (U,\eta,\zeta;A,I,R)=(0,\times,\times;1,0,0) : 2 2 2 similarity classes

No.	S	$e_0$	$t$	$u$	$v$	$m$	$n$	$s_2$	$s_4$	$V_L$	$V_M$	$V_N$	$E$	$𝐌$	$\left|𝐌\right|$
1	1a	$6$	$0$	$0$	$0$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$5$	$\mathrm{𝟓}$	$1$
2	2	$0$	$1$	$1$	$0$	$1$	$1$	$0$	$0$	$0$	$0$	$0$	$5$	$\mathrm{𝟕}$	$18$

Equations

$$E=5+V_N-4\cdot V_L,E^{+}=2+V_M-2\cdot V_L.$$

$$f^4=5^{e_0}\cdot q_1^4\cdots q_t^4\text{ with }{e}_0\in \{0,2,6\},t\ge 0,n=t-s_2-s_4,$$

$$(U_K:N_{N/K}(U_N))=5^U\text{ and }U+1=A+I+R,$$

$$(U_N:U_0)=5^E\text{ and }{V}_N=4\cdot V_L+E-5.$$

$$[5],\text{ species 1a, }(e_0;t,u,v,m;n,s_2,s_4)=(6;0,0,0,1;0,0,0),$$

$$\text{type ϑ, }(U,\eta ,\zeta ;A,I,R)=(0,\times ,\times ;1,0,0),\text{ and }(V_L,V_M,V_N;E)=(0,0,0;5).$$

$$[7],\text{ species 2, }(e_0;t,u,v,m;n,s_2,s_4)=(0;1,1,0,1;1,0,0),$$

$$\text{type ϑ, }(U,\eta ,\zeta ;A,I,R)=(0,\times ,\times ;1,0,0),\text{ and }(V_L,V_M,V_N;E)=(0,0,0;5);$$

$$[2],\text{ species 1b, }(e_0;t,u,v,m;n,s_2,s_4)=(2;1,0,1,1;1,0,0),$$

$$\text{type ε, }(U,\eta ,\zeta ;A,I,R)=(1,\times ,-;2,0,0),\text{ and }(V_L,V_M,V_N;E)=(0,0,0;5);$$

$$[10],\text{ species 1a, }(e_0;t,u,v,m;n,s_2,s_4)=(6;1,0,1,4;1,0,0),$$

$$\text{type ε, }(U,\eta ,\zeta ;A,I,R)=(1,\times ,-;2,0,0),\text{ and }(V_L,V_M,V_N;E)=(0,0,0;5);$$

$$[18],\text{ species 2, }(e_0;t,u,v,m;n,s_2,s_4)=(0;2,0,2,1;2,0,0),$$

$$\text{type ε, }(U,\eta ,\zeta ;A,I,R)=(1,\times ,-;2,0,0),\text{ and }(V_L,V_M,V_N;E)=(0,0,0;5).$$

$$[35],\text{ species 1a, }(e_0;t,u,v,m;n,s_2,s_4)=(6;1,1,0,4;1,0,0),$$

$$\text{type η, }(U,\eta ,\zeta ;A,I,R)=(1,-,\times ;2,0,0),\text{ and }(V_L,V_M,V_N;E)=(0,0,1;6);$$

$$[785],\text{ species 1a, }(e_0;t,u,v,m;n,s_2,s_4)=(6;1,1,0,4;1,0,0),$$

$$\text{type ε, }(U,\eta ,\zeta ;A,I,R)=(1,\times ,-;2,0,0),\text{ and }(V_L,V_M,V_N;E)=(1,2,4;5).$$

$$[301],\text{ species 2, }(e_0;t,u,v,m;n,s_2,s_4)=(0;2,2,0,4;2,0,0),$$

$$\text{type η, }(U,\eta ,\zeta ;A,I,R)=(1,-,\times ;2,0,0),\text{ and }(V_L,V_M,V_N;E)=(0,0,1;6);$$

$$[749],\text{ species 2, }(e_0;t,u,v,m;n,s_2,s_4)=(0;2,2,0,4;2,0,0),$$

$$\text{type ε, }(U,\eta ,\zeta ;A,I,R)=(1,\times ,-;2,0,0),\text{ and }(V_L,V_M,V_N;E)=(1,2,4;5).$$

$$(30,60,90,120,150,180,240,270,360,540,600,720,810,1350,1620,3750)=$$

$$=(2\cdot 3\cdot 5,2^2\cdot 3\cdot 5,2\cdot 3^2\cdot 5,2^3\cdot 3\cdot 5,2\cdot 3\cdot 5^2,2^2\cdot 3^2\cdot 5,2^4\cdot 3\cdot 5,2\cdot 3^3\cdot 5,$$

$$2^3\cdot 3^2\cdot 5,2^2\cdot 3^3\cdot 5,2^3\cdot 3\cdot 5^2,2^4\cdot 3^2\cdot 5,2\cdot 3^4\cdot 5,2\cdot 3^3\cdot 5^2,2^2\cdot 3^4\cdot 5,2\cdot 3\cdot 5^4)$$

$$({\epsilon }^3,{\gamma }^{13})=(\gamma ,\gamma ,\gamma ,\gamma ,\gamma ,\epsilon ,\epsilon ,\gamma ,\gamma ,\gamma ,\gamma ,\gamma ,\gamma ,\epsilon ,\gamma ,\gamma ).$$

$$[30],\text{ species 1a, }(e_0;t,u,v,m;n,s_2,s_4)=(6;2,0,2,16;2,0,0),$$

$$\text{type γ, }(U,\eta ,\zeta ;A,I,R)=(2,-,-;3,0,0),\text{ and }(V_L,V_M,V_N;E)=(0,0,1;6);$$

$$[180],\text{ species 1a, }(e_0;t,u,v,m;n,s_2,s_4)=(6;2,0,2,16;2,0,0),$$

$$\text{type ε, }(U,\eta ,\zeta ;A,I,R)=(1,\times ,-;2,0,0),\text{ and }(V_L,V_M,V_N;E)=(1,2,4;5).$$

$$[70],\text{ species 1a, }(e_0;t,u,v,m;n,s_2,s_4)=(6;2,1,1,16;2,0,0),$$

$$\text{type γ, }(U,\eta ,\zeta ;A,I,R)=(2,-,-;3,0,0),\text{ and }(V_L,V_M,V_N;E)=(0,0,1;6);$$

$$[140],\text{ species 1a, }(e_0;t,u,v,m;n,s_2,s_4)=(6;2,1,1,16;2,0,0),$$

$$\text{type ε, }(U,\eta ,\zeta ;A,I,R)=(1,\times ,-;2,0,0),\text{ and }(V_L,V_M,V_N;E)=(1,2,4;5).$$

$$1\le A\le$$

$$0\le I\le$$

$$0\le R\le$$

Full text

Research supported by the Austrian Science Fund (FWF): projects J 0497-PHY and P 26008-N25

1. Introduction

At the end of his $1975$ article on class numbers of pure quintic fields, Parry suggested verbatim “In conclusion the author would like to say that he believes a numerical study of pure quintic fields would be most interesting” [18, p. 484]. Of course, it would have been rather difficult to realize Parry’s desire in $1975$. But now, $40$ years later, we are in the position to use the powerful computer algebra systems PARI/GP [17] and MAGMA [5, 6, 12] for starting an attack against this hard problem. Prepared by [15, 16], this will actually be done in the present paper.

Even in $1991$, when we generalized Barrucand and Cohn’s theory [4] of principal factorization types from pure cubic fields $\mathbb{Q}(\sqrt[3]{D})$ to pure quintic fields $L=\mathbb{Q}(\sqrt[5]{D})$ and their pure metacyclic normal closures $N=\mathbb{Q}(\zeta_{5},\sqrt[5]{D})$ [13], it was still impossible to verify our hypothesis about the distinction between absolute, intermediate and relative differential principal factors (DPF) [15, (6.3)] and about the values of the unit norm index $(U_{K}:N_{N/K}(U_{N}))$ [15, (1.3)] by actual computations.

All these conjectures have been proven by our most recent numerical investigations. Our classification is based on the Hasse-Iwasawa theorem about the Herbrand quotient of the unit group $U_{N}$ of the Galois closure $N$ of $L$ as a module over the relative group $G=\mathrm{Gal}(N/K)$ with respect to the cyclotomic subfield $K=\mathbb{Q}(\zeta_{5})$. It only involves the unit norm index $(U_{K}:N_{N/K}(U_{N}))$ and our $13$ types of differential principal factorizations [15, Thm. 1.3], but not the index of subfield units $(U_{N}:U_{0})$ [15, § 5] in Parry’s class number formula [15, (5.1)].

We begin with a collection of explicit multiplicity formulas in § 2 which are required for understanding the subsequent extensive presentation of our computational results in twenty tables of crucial invariants in § 3. This information admits the classification of all $900$ pure quintic fields $L=\mathbb{Q}(\root 5 \of{D})$ with normalized radicands $2\leq D<10^{3}$ into $13$ DPF types and the refined classification into similarity classes with representative prototypes in § 4.

In these final conclusions, we collect theoretical consequences of our experimental results and draw the attention to remaining open questions.

2. Collection of Multiplicity Formulas

For the convenience of the reader, we provide a summary of formulas for calculating invariants of pure quintic fields $L=\mathbb{Q}(\sqrt[5]{D})$ with normalized fifth power free radicands $D>1$ and their associated pure metacyclic normal fields $N=\mathbb{Q}(\zeta,\sqrt[5]{D})$ with a primitive fifth root of unity $\zeta=\zeta_{5}$.

Let $f$ be the class field theoretic conductor of the relatively quintic Kummer extension $N/K$ over the cyclotomic field $K=\mathbb{Q}(\zeta)$. It is also called the conductor of the pure quintic field $L$. The multiplicity $m=m(f)$ of the conductor $f$ indicates the number of non-isomorphic pure metacyclic fields $N$ sharing the common conductor $f$, or also, according to [15, Prop. 2.1], the number of normalized fifth power free radicands $D>1$ whose fifth roots generate non-isomorphic pure quintic fields $L$ sharing the common conductor $f$.

We adapt the general multiplicity formulas in [14, Thm. 2, p. 104] to the quintic case $p=5$. If $L$ is a field of species $1\mathrm{a}$ [15, (2.6) and Exm. 2.2], i.e. $f^{4}=5^{6}\cdot q_{1}^{4}\cdots q_{t}^{4}$, then $m=4^{t}$ where $t:=\#\{q\in\mathbb{P}\mid q\neq 5,\ q\mid f\}$. The explicit values of $m$ in dependence on $t$ are given in Table 1.

If $L$ is a field of species $1\mathrm{b}$ [15, (2.6) and Exm. 2.2], i.e. $f^{4}=5^{2}\cdot q_{1}^{4}\cdots q_{t}^{4}$, then $m=4^{u}\cdot X_{v}$ where $u:=\#\{q\in\mathbb{P}\mid q\equiv\pm 1,\pm 7\,(\mathrm{mod}\,25),\ q\mid f\}$, $v:=t-u$ and $X_{j}:=\frac{1}{5}[4^{j}-(-1)^{j}]$, that is $(X_{j})_{j\geq-1}=(\frac{1}{4},0,1,3,13,51,205,\ldots)$. The explicit values of $m$ in dependence on $u$ and $v$ are given in Table 2.

If $L$ is a field of species $2$ [15, (2.6) and Exm. 2.2], i.e. $f^{4}=5^{0}\cdot q_{1}^{4}\cdots q_{t}^{4}$, then $m=4^{u}\cdot X_{v-1}$ where $u:=\#\{q\in\mathbb{P}\mid q\equiv\pm 1,\pm 7\,(\mathrm{mod}\,25),\ q\mid f\}$, $v:=t-u$ and $X_{j}:=\frac{1}{5}[4^{j}-(-1)^{j}]$, that is $(X_{j})_{j\geq-1}=(\frac{1}{4},0,1,3,13,51,205,\ldots)$. The explicit values of $m$ in dependence on $u$ and $v$ are given in Table 3.

3. Classification by DPF types in $20$ numerical tables

3.1. DPF types

The following twenty Tables 6 – 25 establish a complete classification of all $900$ pure metacyclic fields $N=\mathbb{Q}(\zeta,\root 5 \of{D})$ with normalized radicands in the range $2\leq D\leq 10^{3}$. With the aid of PARI/GP [17] and MAGMA [12] we have determined the differential principal factorization type, T, of each field $N$ by means of other invariants $U,A,I,R$ [15, Thm. 6.1]. After several weeks of CPU time, the date of completion was September $17$, $2018$.

The possible DPF types are listed in dependence on $U,A,I,R$ in Table 4, where the symbol $\times$ in the column $\eta$, resp. $\zeta$, indicates the existence of a unit $H\in U_{N}$, resp. $Z\in U_{N}$, such that $\eta=N_{N/K}(H)$, resp. $\zeta=N_{N/K}(Z)$. The $5$-valuation of the unit norm index $(U_{K}:N_{N/K}{U_{N}})$ is abbreviated by $U$ [15, (1.3), (6.3)].

3.2. Justification of the computational techniques

The steps of the following classification algorithm are ordered by increasing requirements of CPU time. To avoid unnecessary time consumption, the algorithm stops at early stages already, as soon as the DPF type is determined unambiguously. The illustrating subfield lattice of $N$ is drawn in Figure 1 at the end of the paper.

Algorithm 3.1**.**

(Classification into $13$ DPF types.)

Input: a normalized fifth power free radicand $D\geq 2$.

Step 1: By purely rational methods, without any number field constructions, the prime factorization of the radicand $D$ (including the counters $t,u,v;n,s_{2},s_{4}$, § 4.2) is determined. If $D=q\in\mathbb{P}$, $q\equiv\pm 2\,(\mathrm{mod}\,5)$, $q\not\equiv\pm 7\,(\mathrm{mod}\,25)$, then $N$ is a Polya field of type $\varepsilon$; stop. If $D=q\in\mathbb{P}$, $q=5$ or $q\equiv\pm 7\,(\mathrm{mod}\,25)$, then $N$ is a Polya field of type $\vartheta$; stop.

Step 2: The field $L$ of degree $5$ is constructed. The primes $q_{1},\ldots,q_{T}$ dividing the conductor $f$ of $N/K$ are determined, and their overlying prime ideals $\mathfrak{q}_{1},\ldots,\mathfrak{q}_{T}$ in $L$ are computed. By means of at most $5^{T}$ principal ideal tests of the elements of $\mathcal{I}_{L/\mathbb{Q}}/\mathcal{I}_{\mathbb{Q}}=\bigoplus_{i=1}^{T}\,\mathbb{F}_{5}\,\mathfrak{q}_{i}$, the number $5^{A}:=\#\{(v_{1},\ldots,v_{T})\in\mathbb{F}_{5}^{T}\mid\prod_{i=1}^{T}\,\mathfrak{q}_{i}^{v_{i}}\in\mathcal{P}_{L}\}$, that is the cardinality of $\mathcal{P}_{L/\mathbb{Q}}/\mathcal{P}_{\mathbb{Q}}$, is determined. If $A=T$, then $N$ is a Polya field. If $A=3$, then $N$ is of type $\gamma$; stop. If $A=2$, $s_{2}=s_{4}=0$, $v\geq 1$, then $N$ is of type $\varepsilon$; stop. If $A=1$, $s_{2}=s_{4}=0$, then $N$ is of type $\vartheta$; stop.

Step 3: If $s_{2}\geq 1$ or $s_{4}\geq 1$, then the field $M$ of degree $10$ is constructed. For the $2$-split primes $\ell_{1},\ldots,\ell_{s_{2}+s_{4}}\equiv\pm 1\,(\mathrm{mod}\,5)$ among the primes $q_{1},\ldots,q_{T}$ dividing the conductor $f$ of $N/K$, the overlying prime ideals $\mathcal{L}_{1},\mathcal{L}_{1}^{\tau},\ldots,\mathcal{L}_{s_{2}+s_{4}},\mathcal{L}_{s_{2}+s_{4}}^{\tau}$ in $M$ are computed. By means of at most $5^{s_{2}+s_{4}}$ principal ideal tests of the elements of $(\mathcal{I}_{M/K^{+}}/\mathcal{I}_{K^{+}})\bigcap\ker(N_{M/L})=\bigoplus_{i=1}^{s_{2}+s_{4}}\,\mathbb{F}_{5}\,\mathcal{K}_{(\ell_{i})}$, where $\mathcal{K}_{(\ell_{i})}=\mathcal{L}_{i}^{1+4\tau}$ for $1\leq i\leq s_{2}+s_{4}$, the number $5^{I}:=\#\{(v_{1},\ldots,v_{s_{2}+s_{4}})\in\mathbb{F}_{5}^{s_{2}+s_{4}}\mid\prod_{i=1}^{s_{2}+s_{4}}\,\mathcal{K}_{(\ell_{i})}^{v_{i}}\in\mathcal{P}_{M}\}$, that is the cardinality of $(\mathcal{P}_{M/K^{+}}/\mathcal{P}_{K^{+}})\bigcap\ker(N_{M/L})$, is determined. If $I=2$, then $N$ is of type $\alpha_{3}$; stop. If $I=1$, $A=2$, then $N$ is of type $\beta_{2}$; stop.

Step 4: If $s_{4}\geq 1$, then the field $N$ of degree $20$ is constructed. For all $4$-split primes $\ell_{s_{2}+1},\ldots,\ell_{s_{2}+s_{4}}$ $\equiv+1\,(\mathrm{mod}\,5)$ among the primes $q_{1},\ldots,q_{T}$ dividing the conductor $f$ of $N/K$, the overlying prime ideals $\mathfrak{L}_{s_{2}+1},\mathfrak{L}_{s_{2}+1}^{\tau^{2}},\mathfrak{L}_{s_{2}+1}^{\tau},\mathfrak{L}_{s_{2}+1}^{\tau^{3}},\ldots,\mathfrak{L}_{s_{2}+s_{4}},\mathfrak{L}_{s_{2}+s_{4}}^{\tau^{2}},\mathfrak{L}_{s_{2}+s_{4}}^{\tau},\mathfrak{L}_{s_{2}+s_{4}}^{\tau^{3}}$ in $N$ are computed. By means of at most $5^{2s_{4}}$ principal ideal tests of the elements of $(\mathcal{I}_{N/K}/\mathcal{I}_{K})\bigcap\ker(N_{N/M})=\bigoplus_{i=s_{2}+1}^{s_{2}+s_{4}}\,\left(\mathbb{F}_{5}\,\mathfrak{K}_{1,(\ell_{i})}\right.$ $\left.\oplus\mathbb{F}_{5}\,\mathfrak{K}_{2,(\ell_{i})}\right)$, where $\mathfrak{K}_{1,(\ell_{i})}=\mathfrak{L}_{i}^{1+4\tau^{2}+2\tau+3\tau^{3}}$ and $\mathfrak{K}_{2,(\ell_{i})}=\mathfrak{L}_{i}^{1+4\tau^{2}+3\tau+2\tau^{3}}$ for $s_{2}+1\leq i\leq s_{2}+s_{4}$, the number $5^{R}:=\#\{(v_{1,s_{2}+1},v_{2,s_{2}+1},\ldots,v_{1,s_{2}+s_{4}},v_{2,s_{2}+s_{4}})\in\mathbb{F}_{5}^{2s_{4}}\mid\prod_{i=s_{2}+1}^{s_{2}+s_{4}}\,\left(\mathfrak{K}_{1,(\ell_{i})}^{v_{1,i}}\mathfrak{K}_{2,(\ell_{i})}^{v_{2,i}}\right)\in\mathcal{P}_{N}\}$, that is the cardinality of $(\mathcal{P}_{N/K}/\mathcal{P}_{K})\bigcap\ker(N_{N/M})$, is determined. If $R=2$, then $N$ is of type $\alpha_{1}$; stop. If $R=1$, $I=1$, then $N$ is of type $\alpha_{2}$; stop. If $R=1$, $A=2$, then $N$ is of type $\beta_{1}$; stop.

Step 5: If the type of the field $N$ is not yet determined uniquely, then $U=1$ and there remain the following possibilities. If $v\geq 1$, then $N$ is of type $\delta_{1}$, if $R=1$, of type $\delta_{2}$, if $I=1$, and of type $\varepsilon$, if $R=I=0$. If $v=0$, then a fundamental system $(E_{j})_{1\leq j\leq 9}$ of units is constructed for the unit group $U_{N}$ of the field $N$ of degree $20$, and all relative norms of these units with respect to the cyclotomic subfield $K$ are computed. If $N_{N/K}(E_{j})=\zeta_{5}^{k}$ for some $1\leq j\leq 9$, $1\leq k\leq 4$, then $N$ is of type $\zeta_{1}$, if $R=1$, of type $\zeta_{2}$, if $I=1$, and of type $\eta$, if $R=I=0$. Otherwise the conclusions are the same as for $v\geq 1$.

Output: the DPF type of the field $N=\mathbb{Q}(\zeta_{5},\sqrt[5]{D})$ and the decision about its Polya property.

Proof.

The claims of Step 1 concerning the types $\varepsilon,\vartheta$ are proved in items (1) and (2) of [15, Thm. 10.1].

For Step 2, the formulas (4.1) and (4.2) in [15, Thm. 4.1] give an $\mathbb{F}_{5}$-basis of the space of absolute differential factors, and the formulas (4.3) and (4.4) in [15, Cor. 4.1] determine bounds for the $\mathbb{F}_{5}$-dimension $A$ of the space of absolute DPF in the field $L$ of degree $5$. The Polya property was characterized in [15, Thm. 10.5], the claim concerning type $\gamma$ follows from [15, Thm. 6.1], and the claims about the types $\varepsilon,\vartheta$ from [15, Thm. 8.1 and Thm. 6.1].

For Step 3, the formulas (4.5) and (4.6) in [15, Thm. 4.3] give an $\mathbb{F}_{5}$-basis of the space of intermediate differential factors, and the formulas (4.7) and (4.8) in [15, Cor. 4.2] determine bounds for the $\mathbb{F}_{5}$-dimension $I$ of the space of intermediate DPF in the field $M$ of degree $10$. The claims concerning the types $\alpha_{3},\beta_{2}$ are consequences of [15, Thm. 6.1],

For Step 4, the formulas (4.9) and (4.10) in [15, Thm. 4.4] give an $\mathbb{F}_{5}$-basis of the space of relative differential factors, and the formulas (4.11) and (4.12) in [15, Cor. 4.3] determine bounds for the $\mathbb{F}_{5}$-dimension $R$ of the space of relative DPF in the field $N$ of degree $20$. The claims concerning the types $\alpha_{1},\alpha_{2},\beta_{1}$ are consequences of [15, Thm. 6.1].

Concerning Step 5, the signature of $N$ is $(r_{1},r_{2})=(0,10)$, whence the torsion free Dirichlet unit rank of $N$ is given by $r=r_{1}+r_{2}-1=9$. The claims about all types are consequences of [15, Thm. 6.1], including information on the constitution of the norm group $N_{N/K}(U_{N})$. ∎

Remark 3.1**.**

Whereas the execution of Step 1 and 2 in our Algorithm 3.1, implemented as a Magma program script [12], is a matter of a few seconds on a machine with clock frequency at least $2\,$GHz, the CPU time for Step 3 lies in the range of several minutes. The time requirement for Step 4 and 5 can reach hours or even days in spite of code optimizations for the calculation of units, in particular the use of the Magma procedures IndependentUnits() and SetOrderUnitsAreFundamental() prior to the call of UnitGroup().

3.3. Open problems

We conjecture that considerable amounts of CPU time can be saved in our Algorithm 3.1 by computing the logarithmic $5$-class numbers $V_{F}:=v_{5}(h_{F})$ of the fields $F\in\{L,M,N\}$, which admit the determination of the logarithmic indices $E$, resp. $E^{+}$, of subfield units in the Parry [18], resp. Kobayashi [10, 11], class number relation, according to the formulas

[TABLE]

However, first there would be required rigorous proofs of the heuristic connections between $E,E^{+}$ and the DPF types in Table 5, where $(E,E^{+})=(1,0)$ implies type $\alpha_{2}$, $(E,E^{+})=(2,0)$ implies type $\alpha_{3}$, $(E,E^{+})=(4,2)$ implies type $\delta_{1}$, but $(E,E^{+})=(2,1)$ admits types $\alpha_{1},\delta_{1}$, $(E,E^{+})=(3,1)$ admits types $\alpha_{2},\beta_{1},\delta_{2}$, $(E,E^{+})=(4,1)$ admits types $\beta_{2},\zeta_{2}$, $(E,E^{+})=(5,2)$ admits types $\beta_{1},\varepsilon,\zeta_{1},\vartheta$, and $(E,E^{+})=(6,2)$ admits types $\gamma,\eta$. $(E,E^{+})=(0,0)$ seems to be impossible.

3.4. Conventions and notation in the tables

The normalized radicand $D=q_{1}^{e_{1}}\cdots q_{s}^{e_{s}}$ of a pure metacyclic field $N$ of degree $20$ is minimal among the powers $D^{n}$, $1\leq n\leq 4$, with corresponding exponents $e_{j}$ reduced modulo $5$. The normalization of the radicands $D$ provides a warranty that all fields are pairwise non-isomorphic [15, Prop. 2.1].

Prime factors are given for composite radicands $D$ only. Dedekind’s species, S, of radicands is refined by distinguishing $5\mid D$ (species 1a) and $\gcd(5,D)=1$ (species 1b) among radicands $D\not\equiv\pm 1,\pm 7\,(\mathrm{mod}\,25)$ (species 1). By the species and factorization of $D$, the shape of the conductor $f$ is determined. We give the fourth power $f^{4}$ to avoid fractional exponents. Additionally, the multiplicity $m$ indicates the number of non-isomorphic fields sharing a common conductor $f$ (§2). The symbol $V_{F}$ briefly denotes the $5$-valuation of the order $h_{F}=\#\mathrm{Cl}(F)$ of the class group $\mathrm{Cl}(F)$ of a number field $F$. By $E$ we denote the exponent of the power in the index of subfield units $(U_{N}:U_{0})=5^{E}$.

An asterisk denotes the smallest radicand with given Dedekind kind, DPF type and $5$-class groups $\mathrm{Cl}_{5}(F)$, $F\in\{L,M,N\}$. The latter are usually elementary abelian, except for the cases indicated by an additional asterisk (see § 4.4).

Principal factors, P, are listed when their constitution is not a consequence of the other information. According to [15, Thm. 7.2., item (1)] it suffices to give the rational integer norm of absolute principal factors. For intermediate principal factors, we use the symbols $\mathcal{K}:=\mathcal{L}^{1-\tau}=\alpha\mathcal{O}_{M}$ with $\alpha\in M$ or $\mathcal{L}=\lambda\mathcal{O}_{M}$ with a prime element $\lambda\in M$ (which implies $\mathcal{L}^{\tau}=\lambda^{\tau}\mathcal{O}_{M}$ and thus also $\mathcal{K}=\lambda^{1-\tau}\mathcal{O}_{M}$). Here, $(\mathcal{L}^{1+\tau})^{5}=\ell\mathcal{O}_{M}$ when a prime $\ell\equiv\pm 1\,(\mathrm{mod}\,5)$ divides the radicand $D$. For relative principal factors, we use the symbols $\mathfrak{K}_{1}:=\mathfrak{L}^{1+4\tau^{2}+2\tau+3\tau^{3}}=A_{1}\mathcal{O}_{N}$ and $\mathfrak{K}_{2}:=\mathfrak{L}^{1+4\tau^{2}+3\tau+2\tau^{3}}=A_{2}\mathcal{O}_{N}$ with $A_{1},A_{2}\in N$. Here, $(\mathfrak{L}^{1+\tau+\tau^{2}+\tau^{3}})^{5}=\ell\mathcal{O}_{N}$ when a prime number $\ell\equiv+1\,(\mathrm{mod}\,5)$ divides the radicand $D$. (Kernel ideals in [15, § 7].)

The quartet $(1,2,4,5)$ indicates conditions which either enforce a reduction of possible DPF types or enable certain DPF types. The lack of a prime divisor $\ell\equiv\pm 1\,(\mathrm{mod}\,5)$ together with the existence of a prime divisor $q\not\equiv\pm 7\,(\mathrm{mod}\,25)$ and $q\neq 5$ of $D$ is indicated by a symbol $\times$ for the component $1$. In these cases, only the two DPF types $\gamma$ and $\varepsilon$ can occur [15, Thm. 8.1].

A symbol $\times$ for the component $2$ emphasizes a prime divisor $\ell\equiv-1\,(\mathrm{mod}\,5)$ of $D$ and the possibility of intermediate principal factors in $M$, like $\mathcal{L}$ and $\mathcal{K}$. A symbol $\times$ for the component $4$ emphasizes a prime divisor $\ell\equiv+1\,(\mathrm{mod}\,5)$ of $D$ and the possibility of relative principal factors in $N$, like $\mathfrak{K}_{1}$ and $\mathfrak{K}_{2}$. The $\times$ symbol is replaced by $\otimes$ if the facility is used completely, and by $(\times)$ if the facility is only used partially.

If $D$ has only prime divisors $q\equiv\pm 1,\pm 7\,(\mathrm{mod}\,25)$ or $q=5$, a symbol $\times$ is placed in component $5$. In these cases, $\zeta$ can occur as a norm $N_{N/K}(Z)$ of some unit in $Z\in U_{N}$. If it actually does, the $\times$ is replaced by $\otimes$ [15, § 8].

4. Statistical evaluation, refinements, and conclusions

4.1. Statistics of DPF types

The complete statistical evaluation of the preceding twenty Tables 6 — 25 is given in Table 26. The first ten columns show the absolute frequencies of pure metacyclic fields $N=\mathbb{Q}(\zeta,\sqrt[5]{D})$ with various DPF types for the ranges $2\leq D<n\cdot 100$ with $1\leq n\leq 10$. The eleventh column lists the relative percentages of the five most frequent DPF types for the complete range $2\leq D<10^{3}$ of normalized radicands.

Among our $13$ differential principal factorization types, type $\gamma$ with $3$-dimensional absolute principal factorization, $A=3$, is clearly dominating with more than one third $(36\,\%)$ of all occurrences in the complete range $2\leq D<10^{3}$, followed by type $\varepsilon$ with $2$-dimensional absolute principal factorization, $A=2$, which covers nearly one quarter $(23\,\%)$ of all cases. The third place (nearly $18\,\%$) is occupied by type $\beta_{2}$ with mixed absolute and intermediate principal factorization, $A=2$, $I=1$.

It is striking that type $\alpha_{1}$ with $2$-dimensional relative principal factorization, $R=2$, and type $\alpha_{3}$ with $2$-dimensional intermediate principal factorization, $I=2$, are populated rather sparsely, in favour of a remarkable contribution by type $\alpha_{2}$ with mixed intermediate and relative principal factorization, $I=R=1$ (place four with $8\,\%$).

The appearance of the four types $\zeta_{1}$, $\zeta_{2}$, $\eta$, $\vartheta$ with norm representation $N_{N/K}(Z)=\zeta$, $Z\in U_{N}$, of the primitive fifth root of unity $\zeta=\zeta_{5}$ is marginal ([15, Thm. 8.2]), in spite of the parametrized contribution by all prime conductors $f=q\equiv\pm 7\,(\mathrm{mod}\,25)$ to type $\vartheta$, as we shall prove in Theorem 4.1 (1) in § 4.3.

4.2. Similarity classes and prototypes

In [16], we came to the conviction that for deeper insight into the arithmetical structure of the fields under investigation, the prime factorization of the class field theoretic conductor $f$ of the abelian extension $N/K$ over the cyclotomic field $K=\mathbb{Q}(\zeta)$ and the primary invariants of all involved $5$-class groups must be taken in consideration. These ideas have lead to the concept of similarity classes and representative prototypes, which refines the differential principal factorization (DPF) types

Let $t$ be the number of primes $q_{1},\ldots,q_{t}\in\mathbb{P}$ distinct from $5$ which divide the conductor $f$. Among these prime numbers, we separately count $u:=\#\{1\leq i\leq t\mid q_{i}\equiv\pm 1,\pm 7\,(\mathrm{mod}\,25)\}$ free primes, $v:=t-u$ restrictive primes, $s_{2}:=\#\{1\leq i\leq t\mid q_{i}\equiv-1\,(\mathrm{mod}\,5)\}$ $2$-split primes, and $s_{4}:=\#\{1\leq i\leq t\mid q_{i}\equiv+1\,(\mathrm{mod}\,5)\}$ $4$-split primes. The multiplicity $m=m(f)$ is given in terms of $t,u,v$, according to § 2, and the dimensions of various spaces of primitive ambiguous ideals over the finite field $\mathbb{F}_{5}$ are given in terms of $t,s_{2},s_{4}$, according to [15, § 4]. By $\eta=\frac{1}{2}(1+\sqrt{5})$ we denote the fundamental unit of $K^{+}=\mathbb{Q}(\sqrt{5})$. The dimensions of the spaces of absolute, intermediate and relative DPF over $\mathbb{F}_{5}$ are denoted by $A$, $I$ and $R$, identical with the additive (logarithmic) version in [15, Thm. 6.1]. Further, let $M=\mathbb{Q}(\sqrt{5},\sqrt[5]{D})$ be the maximal real intermediate field of $N/L$, and denote by $U_{0}$ the subgroup of the unit group $U_{N}$ of $N=\mathbb{Q}(\zeta,\sqrt[5]{D})$ generated by the units of all conjugate fields of $L=\mathbb{Q}(\sqrt[5]{D})$ and of $K=\mathbb{Q}(\zeta)$, where $\zeta=\zeta_{5}$ is a primitive fifth root of unity. For a number field $F$, let $V_{F}:=v_{5}(\#\mathrm{Cl}(F))$ be the $5$-valuation of the class number of $F$.

Definition 4.1**.**

A set of normalized fifth power free radicands $D>1$ is called a similarity class if the associated pure quintic fields $L=\mathbb{Q}(\sqrt[5]{D})$ share the following common multiplets of invariants:

• •

the refined Dedekind species $(e_{0};t,u,v,m;n,s_{2},s_{4})$, where

[TABLE]

• •

the differential principal factorization type $(U,\eta,\zeta;A,I,R)$, where

[TABLE]

• •

the structure of the $5$-class groups $(V_{L},V_{M},V_{N};E)$, where

[TABLE]

Warning 4.1**.**

To reduce the number of invariants, we abstain from defining additional counters $s_{2}^{\prime}:=\#\{1\leq i\leq t\mid q_{i}\equiv-1\,(\mathrm{mod}\,25)\}$ and $s_{4}^{\prime}:=\#\{1\leq i\leq t\mid q_{i}\equiv+1\,(\mathrm{mod}\,25)\}$ for free splitting prime divisors of the conductor $f$. However, we point out that occasionally a similarity class in the sense of Definition 4.1 will be split in two separate classes, having the same invariants, but distinct contributions to the counters $u$ and $s_{4}$, resp. $s_{2}$. For instance, the similarity classes $\mathbf{[77]}$ and $\mathbf{[202]}$ with $77=7\cdot 11$ and $202=2\cdot 101$ share identical multiplets of invariants $(e_{0};t,u,v,m;n,s_{2},s_{4})=(2;2,1,1,4;1,0,1)$ (species $1\mathrm{b}$), $(U,\eta,\zeta;A,I,R)=(2,-,-;2,1,0)$ (type $\beta_{2}$), and $(V_{L},V_{M},V_{N};E)=(1,1,3;4)$. But $u=1$ and $n=1$ are due to $7$, $v=1$ and $s_{4}=1$ are due to $11$, in the former case, whereas $v=1$ and $n=1$ are due to $2$, $u=1$ and $s_{4}=1$ are due to $101$, in the latter case. Therefore, the contributions by primes congruent to $\pm 1\,(\mathrm{mod}\,25)$ will be indicated by writing $u=1^{\prime}$ and $s_{4}=1^{\prime}$, resp. $s_{2}=1^{\prime}$.

We also emphasize that in the rare cases of non-elementary $5$-class groups, the actual structures (abelian type invariants) of the $5$-class groups will be taken into account, and not only the $5$-valuations $V_{L},V_{M},V_{N}$.

Definition 4.2**.**

The minimal element $\mathbf{M}$ of a similarity class (with respect to the natural order of positive integers $\mathbb{N}$) is called the representative prototype of the class, which is denoted by writing its prototype in square brackets $\mathbf{[M]}$.

The remaining elements of a similarity class, which are bigger than the prototype, only reproduce the arithmetical invariants of the prototype and do not provide any additional information, exept possibly about other primary components of the class groups, that is the structure of $\ell$-class groups $\mathrm{Cl}_{\ell}(F)$ of the fields $F\in\{L,M,N\}$ for $\ell\in\mathbb{P}\setminus\{5\}$.

Whereas there are only $13$ DPF types of pure quintic fields, the number of similarity classes is obviously infinite, since firstly the number $t$ of primes dividing the conductor is unbounded and secondly the number of states, defined by the triplet $(V_{L},V_{M},V_{N})$ of $5$-valuations of class numbers, is also unlimited.

Given a fixed refined Dedekind species $(e_{0};t,u,v,m;n,s_{2},s_{4})$, the set of all associated normalized fifth power free radicands $D$ usually splits into several similarity classes defined by distinct DPF types (type splitting). Occasionally it even splits further into different structures of $5$-class groups, called states, with increasing complexity of abelian type invariants (state splitting).

The $134$ prototypes $2\leq\mathbf{M}<10^{3}$ of pure quintic fields are listed in the Tables 27, 28, 29 and 30. By $\mathbf{\lvert M\rvert}:=\#\{D\in\mathbf{[M]}\mid D<B\}$ we denote the number of elements of the similarity class $\mathbf{[M]}$ defined by the prototype $\mathbf{M}$, truncated at the upper bound $B:=10^{3}$ of our systematic investigations.

4.3. General theorems on DPF types and Polya fields

There is only a single finite similarity class $[5]=\{5\}$, characterized by the exceptional number $t=0$ of primes $q\neq 5$ dividing the conductor $f$ (here $f^{4}=5^{6}$). The invariants of this unique metacyclic Polya field $N$ are given by

[TABLE]

We conjecture that all the other similarity classes are infinite. Precisely four of them can actually be given by parametrized infinite sequences in a deterministic way aside from the intrinsic probabilistic nature of the occurrence of primes in residue classes and of composite integers with assigned shape of prime decomposition. This was proved in [15, Thm. 10.1] and [16, Thm. 2.1].

Theorem 4.1**.**

Each of the following infinite sequences of conductors $f=f_{N/K}$ unambiguously determines the DPF type of the pure metacyclic fields $N$ in the associated multiplet with $m=m(f)$ members.

1. (1)

$f=q$* with $q\in\mathbb{P}$, $q\equiv\pm 7\,(\mathrm{mod}\,25)$ gives rise to a singulet, $m=1$, with DPF type $\vartheta$,* 2. (2)

$f^{4}=5^{2}\cdot q^{4}$* with $q\in\mathbb{P}$, $q\equiv\pm 2\,(\mathrm{mod}\,5)$, $q\not\equiv\pm 7\,(\mathrm{mod}\,25)$ gives rise to a singulet, $m=1$, with DPF type $\varepsilon$,* 3. (3)

$f^{4}=5^{6}\cdot q^{4}$* with $q\in\mathbb{P}$, $q\equiv\pm 2\,(\mathrm{mod}\,5)$, $q\not\equiv\pm 7\,(\mathrm{mod}\,25)$ gives rise to a quartet, $m=4$, with homogeneous DPF type $(\varepsilon,\varepsilon,\varepsilon,\varepsilon)$,* 4. (4)

$f=q_{1}\cdot q_{2}$* with $q_{i}\in\mathbb{P}$, $q_{i}\equiv\pm 2\,(\mathrm{mod}\,5)$, $q_{i}\not\equiv\pm 7\,(\mathrm{mod}\,25)$ gives rise to a singulet, $m=1$, with DPF type $\varepsilon$.*

In fact, the shape of the conductors in Theorem 4.1 does not only determine the refined Dedekind species and the DPF type, but also the structure of the $5$-class groups of the fields $L$, $M$ and $N$.

Corollary 4.1**.**

The invariants of the similarity classes defined by the four infinite sequences of conductors in Theorem 4.1 are given as follows, in the same order:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The pure metacyclic fields $N$ associated with these four similarity classes are Polya fields.

Remark 4.1**.**

The statements concerning $5$-class groups in Corollary 4.1 were proved by Parry in [18, Thm. IV, p. 481], where Formula (10) gives the shape of radicands associated with the conductors in our Theorem 4.1.

Proof.

(of Theorem 4.1 and Corollary 4.1) It only remains to show the claims for the composite radicands associated with conductors $f^{4}=5^{6}\cdot q^{4}$ and $f=q_{2}\cdot q_{2}$. See [15, Thm. 10.6]. ∎

For similarity classes distinct from the four infinite classes in Theorem 4.1 we cannot provide deterministic criteria for the DPF type and for the homogeneity of multiplets with $m>1$. In general, the members of a multiplet belong to distinct similarity classes, thus giving rise to heterogeneous DPF types. We explain these phenomena with the simplest cases where only two DPF types are involved (type splitting).

Theorem 4.2**.**

Each of the following infinite sequences of conductors $f=f_{N/K}$ admits precisely two DPF types of the pure metacyclic fields $N$ in the associated quartet with $m=4$ members.

1. (1)

$f^{4}=5^{6}\cdot q^{4}$* with $q\in\mathbb{P}$, $q\equiv\pm 7\,(\mathrm{mod}\,25)$ gives rise to a quartet with possibly heterogeneous DPF type $(\varepsilon^{x},\eta^{y})$, $x+y=4$,* 2. (2)

$f=q_{1}\cdot q_{2}$* with $q_{i}\in\mathbb{P}$, $q_{i}\equiv\pm 7\,(\mathrm{mod}\,25)$ gives rise to a quartet with possibly heterogeneous DPF type $(\varepsilon^{x},\eta^{y})$, $x+y=4$.*

Example 4.1**.**

It is not easy to find complete quartets, whose members are spread rather widely. The smallest quartet $(35,175,245,4375)=(5\cdot 7,5^{2}\cdot 7,5\cdot 7^{2},5^{4}\cdot 7)$ belonging to the first infinite sequence contains the member $D=4375$ outside of the range of our systematic computations. We have determined its DPF type separately and thus discovered a homogeneous quartet of type $(\eta,\eta,\eta,\eta)$. However, we cannot generally exclude the occurrence of heterogeneous quartets.

Corollary 4.2**.**

*The invariants of the similarity classes defined by the two infinite sequences of conductors in Theorem 4.2 are given as follows, in the same order. The statements concerning $5$-class groups are only conjectural. Each sequence splits into two similarity classes.

The classes for $f^{4}=5^{6}\cdot q^{4}$ are:*

[TABLE]

[TABLE]

The classes for $f=q_{1}\cdot q_{2}$ are:

[TABLE]

[TABLE]

All pure metacyclic fields $N$ associated with these four similarity classes are Polya fields.

Proof.

(of Theorem 4.2 and Corollary 4.2) ∎

Remark 4.2**.**

The statements on $5$-class groups in Corollary 4.2 have been verified for all examples with $2\leq D<1000$ by our computations. In particular, the occurrence of the radicands $D=749=7\cdot 107$ and $D=785=5\cdot 157$, both with $V_{L}=1$, proves the impossibility of the general claim $5\nmid h(L)$ for the two situations mentioned in [8, Lem. 3.3 (ii) and (iv), p. 204] and [11, Thm. 5 (ii) and (iv), p. 5], partially also indicated in [18, Thm. IV (11), p. 481].

Theorem 4.3**.**

Each of the following infinite sequences of conductors $f=f_{N/K}$ admits precisely two DPF types of the pure metacyclic fields $N$ in the associated hexadecuplet with $m=16$ members.

1. (1)

$f^{4}=5^{6}\cdot q_{1}^{4}q_{2}^{4}$* with $q_{i}\in\mathbb{P}$, $q_{i}\equiv\pm 2\,(\mathrm{mod}\,5)$, both $q_{i}\not\equiv\pm 7\,(\mathrm{mod}\,25)$ gives rise to a hexadecuplet with possibly heterogeneous DPF type $(\varepsilon^{x},\gamma^{y})$, $x+y=16$,* 2. (2)

$f^{4}=5^{6}\cdot q_{1}^{4}q_{2}^{4}$* with $q_{i}\in\mathbb{P}$, $q_{i}\equiv\pm 2\,(\mathrm{mod}\,5)$, only one $q_{i}\equiv\pm 7\,(\mathrm{mod}\,25)$ gives rise to a hexadecuplet with possibly heterogeneous DPF type $(\varepsilon^{x},\gamma^{y})$, $x+y=16$.*

Example 4.2**.**

It is difficult to find complete hexadecuplets, whose members are spread rather widely. The smallest hexadecuplet

[TABLE]

[TABLE]

[TABLE]

belonging to the first infinite sequence contains the members $D=1350,1620,3750$ outside of the range of our systematic computations. We have determined their DPF type separately and thus discovered a heterogeneous hexadecuplet (in the same order) of type

[TABLE]

Corollary 4.3**.**

*The invariants of the similarity classes defined by the two infinite sequences of conductors in Theorem 4.3 are given as follows, in the same order. The statements concerning $5$-class groups are only conjectural. Each sequence splits into two similarity classes.

The classes for $f^{4}=5^{6}\cdot q_{1}^{4}q_{2}^{4}$, both $q_{i}\not\equiv\pm 7\,(\mathrm{mod}\,25)$ are:*

[TABLE]

[TABLE]

The classes for $f^{4}=5^{6}\cdot q_{1}^{4}q_{2}^{4}$, only one $q_{i}\equiv\pm 7\,(\mathrm{mod}\,25)$ are:

[TABLE]

[TABLE]

Only the pure metacyclic fields $N$ of type $\gamma$ associated with (4.13) and (4.15) are Polya fields.

Proof.

(of Theorem 4.3 and Corollary 4.3) See [15, Thm. 10.7]. ∎

Theorem 4.4**.**

A pure metacyclic field $N=\mathbb{Q}(\zeta_{5},\sqrt[5]{\ell})$ with prime radicand $\ell\equiv\pm 1\,(\mathrm{mod}\,25)$ has a prime conductor $f=\ell$, and possesses the Polya property, regardless of its DPF type and the complexity of its $5$-class group structure.

Proof.

This is an immediate consequence of [15, Thm. 10.5 and Thm. 6.1], taking into account that we have the value $t=1$ for the number of primes dividing the conductor in the present situation, and thus the estimate in [15, Cor. 4.1] yields $1\leq A\leq\min(3,t)=\min(3,1)=1$. For the Polya property we must have $A=t=1$, according to [15, Thm. 10.5], which admits the DPF types $\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$, $\delta_{1}$, $\delta_{2}$, $\zeta_{1}$, $\zeta_{2}$ or $\vartheta$ [15, Thm. 1.3 and Tbl. 1]. However, DPF type $\alpha_{3}$ is excluded by [15, Cor. 4.2], since the requirement $s_{2}+s_{4}\geq 2$ cannot be fulfilled in our situation where either $s_{2}=0$ and $s_{4}=1$ for $\ell\equiv+1\,(\mathrm{mod}\,25)$ or $s_{2}=1$ and $s_{4}=0$ for $\ell\equiv-1\,(\mathrm{mod}\,25)$. ∎

Theorem 4.5**.**

A pure metacyclic field $N=\mathbb{Q}(\zeta_{5},\sqrt[5]{\ell})$ with prime radicand $\ell\equiv\pm 1\,(\mathrm{mod}\,5)$ but $\ell\not\equiv\pm 1\,(\mathrm{mod}\,25)$ has a composite conductor $f^{4}=5^{2}\cdot\ell^{4}$, and the following conditions are equivalent:

1. (1)

$N$* possesses the Polya property.* 2. (2)

$(\exists\,\alpha\in L=\mathbb{Q}(\sqrt[5]{\ell}))\,N_{L/\mathbb{Q}}(\alpha)=5$. 3. (3)

The prime ideal $\mathfrak{p}\in\mathbb{P}_{L}$ with $5\mathcal{O}_{L}=\mathfrak{p}^{5}$ is principal. 4. (4)

$N$* is of DPF type either $\beta_{1}$ or $\beta_{2}$ or $\varepsilon$.*

Proof.

This is a consequence of [15, Thm. 10.5 and Thm. 6.1], taking into account that the prime $5$ is not included in the current definition of the counter $t$ (with value $t=1$ in the present situation), and thus the estimate in [15, Cor. 4.1] must be replaced by $1\leq A\leq\min(3,t+1)=\min(3,2)=2$. For the Polya property we must have $A=t+1=2$ [15, Thm. 10.5], which determines the DPF types $\beta_{1}$, $\beta_{2}$, $\varepsilon$ or $\eta$ [15, Thm. 1.3 and Tbl. 1]. However, DPF type $\eta$ is excluded by the prime $\ell\not\equiv\pm 1,\pm 7\,(\mathrm{mod}\,25)$ dividing the conductor ([15, Thm. 8.1]). ∎

Inspired by the last two theorems, it is worth ones while to summarize, for each kind of prime radicands, what is known about the possibilities for differential principal factorizations.

Theorem 4.6**.**

Let $N=\mathbb{Q}(\zeta_{5},\sqrt[5]{D})$ be a pure metacyclic field with prime radicand $D\in\mathbb{P}$.

1. (1)

If $D=q$ with $q\equiv\pm 7\,(\mathrm{mod}\,25)$ or $q=5$, then $N$ is of type $\vartheta$. 2. (2)

If $D=\ell$ with $\ell\equiv-1\,(\mathrm{mod}\,25)$, then $N$ is of one of the types $\delta_{2},\zeta_{2},\vartheta$. 3. (3)

If $D=\ell$ with $\ell\equiv+1\,(\mathrm{mod}\,25)$, then $N$ is of one of the types $\alpha_{1},\alpha_{2},\delta_{1},\delta_{2},\zeta_{1},\zeta_{2},\vartheta$. 4. (4)

If $D=q$ with $q\equiv\pm 2\,(\mathrm{mod}\,5)$ but $q\not\equiv\pm 7\,(\mathrm{mod}\,25)$, then $N$ is of type $\varepsilon$. 5. (5)

If $D=\ell$ with $\ell\equiv-1\,(\mathrm{mod}\,5)$ but $\ell\not\equiv-1\,(\mathrm{mod}\,25)$, then $N$ is of one of the types $\beta_{2},\delta_{2},\varepsilon$. 6. (6)

If $D=\ell$ with $\ell\equiv+1\,(\mathrm{mod}\,5)$ but $\ell\not\equiv+1\,(\mathrm{mod}\,25)$, then $N$ is of one of the types $\alpha_{1},\alpha_{2},\beta_{1},\beta_{2},\delta_{1},\delta_{2},\varepsilon$.

A pure metacyclic field with prime radicand can never be of any of the types $\alpha_{3},\gamma,\eta$.

Proof.

By making use of the bounds [15, § 4] for $\mathbb{F}_{5}$-dimensions of spaces of differential principal factors (DPF),

[TABLE]

we can determine the possible DPF types of pure metacyclic fields $N=\mathbb{Q}(\zeta_{5},\sqrt[5]{D})$ with prime radicands $D\in\mathbb{P}$. We start with a few general observations.

Firstly, if $D\equiv\pm 1,\pm 7\,(\mathrm{mod}\,25)$, resp. $D=5$, is prime, then $N$ is of Dedekind species $2$, resp. $1\mathrm{a}$, with prime power conductor $f=D$, resp. $f^{4}=5^{6}$, and $t=1$, whence $A=1$ and the types $\beta_{1},\beta_{2},\gamma,\varepsilon,\eta$ with $A\geq 2$ are forbidden. However, if $D\not\equiv\pm 1,\pm 7\,(\mathrm{mod}\,25)$ and $D\neq 5$ is prime, then the congruence requirement eliminates the types $\zeta_{1},\zeta_{2},\eta,\vartheta$, the field $N$ is of Dedekind species $1\mathrm{b}$ with composite conductor $f^{4}=5^{2}\cdot D^{4}$, and $t=2$, whence $1\leq A\leq 2$ and type $\gamma$ with $A=3$ is discouraged. So, the types $\gamma$ and $\eta$ are generally forbidden for prime radicands.

Secondly, for a prime radicand $D\equiv\pm 1\,(\mathrm{mod}\,5)$ which splits in $M$, the space of radicals $\Delta=\langle\sqrt[5]{D}\rangle$ is a $1$-dimensional subspace of absolute DPF contained in the $2$-dimensional space $\Delta\oplus\Delta^{\prime}$ of differential factors generated by the two prime ideals of $M$ over $D$. Consequently, in this special situation there arises an additional constraint $I\leq 1$ for the dimension of the space of intermediate DPF, which must be contained in the $1$-dimensional complement $\Delta^{\prime}$. This generally excludes type $\alpha_{3}$ with $I=2$ for prime radicands.

1. (1)

If $D=q$ with $q\equiv\pm 7\,(\mathrm{mod}\,25)$, then $t=1$, $s_{2}=s_{4}=0$, and thus $A=1$, $I=R=0$. These conditions eliminate the types $\alpha_{1},\alpha_{2},\alpha_{3},\beta_{1},\beta_{2},\gamma,\delta_{1},\delta_{2},\varepsilon,\zeta_{1},\zeta_{2},\eta$ with either $A\geq 2$ or $I\geq 1$ or $R\geq 1$, and only type $\vartheta$ remains admissible. 2. (2)

If $D=\ell$ with $\ell\equiv-1\,(\mathrm{mod}\,25)$, then $t=s_{2}=1$, $s_{4}=0$, and thus $A=1$, $0\leq I\leq 1$, $R=0$, whence the types $\alpha_{1},\alpha_{2},\alpha_{3},\beta_{1},\beta_{2},\gamma,\delta_{1},\varepsilon,\zeta_{1},\eta$ with either $A\geq 2$ or $I=2$ or $R\geq 1$ are excluded, and only the types $\delta_{2},\zeta_{2},\vartheta$ remain admissible. 3. (3)

If $D=\ell$ with $\ell\equiv+1\,(\mathrm{mod}\,25)$, then $t=s_{4}=1$, $s_{2}=0$, and thus $A=1$, $0\leq I\leq 1$, $0\leq R\leq 2$, whence the types $\alpha_{3},\beta_{1},\beta_{2},\gamma,\varepsilon,\eta$ with either $A\geq 2$ or $I=2$ are excluded, and only the types $\alpha_{1},\alpha_{2},\delta_{1},\delta_{2},\zeta_{1},\zeta_{2},\vartheta$ remain admissible. 4. (4)

If $D=q$ with $q\equiv\pm 2\,(\mathrm{mod}\,5)$ but $q\not\equiv\pm 7\,(\mathrm{mod}\,25)$, then $t=2$, $s_{2}=s_{4}=0$, and thus $1\leq A\leq 2$, $I=R=0$. These conditions eliminate the types $\alpha_{1},\alpha_{2},\alpha_{3},\beta_{1},\beta_{2},\gamma,\delta_{1},\delta_{2},\zeta_{1},\zeta_{2}$ with either $A=3$ or $I\geq 1$ or $R\geq 1$, and only the types $\varepsilon,\eta,\vartheta$ remain admissible. However, the congruence requirement modulo $25$ discourages the types $\eta,\vartheta$, and only type $\varepsilon$ is possible. 5. (5)

If $D=\ell$ with $\ell\equiv-1\,(\mathrm{mod}\,5)$ but $\ell\not\equiv-1\,(\mathrm{mod}\,25)$, then $t=2$, $s_{2}=1$, $s_{4}=0$, and thus $1\leq A\leq 2$, $0\leq I\leq 1$, $R=0$, whence the types $\alpha_{1},\alpha_{2},\alpha_{3},\beta_{1},\gamma,\delta_{1},\zeta_{1}$ with either $A=3$ or $I=2$ or $R\geq 1$ are forbidden. The types $\zeta_{2},\eta,\vartheta$ are excluded by congruence conditions, and only the types $\beta_{2},\delta_{2},\varepsilon$ remain admissible. 6. (6)

If $D=\ell$ with $\ell\equiv+1\,(\mathrm{mod}\,5)$ but $\ell\not\equiv+1\,(\mathrm{mod}\,25)$ then $t=2$, $s_{2}=0$, $s_{4}=1$, and thus $1\leq A\leq 2$, $0\leq I\leq 1$, $0\leq R\leq 2$, whence the types $\alpha_{3},\gamma$ with either $A=3$ or $I=2$ are forbidden. The types $\zeta_{1},\zeta_{2},\eta,\vartheta$ are excluded by congruence conditions, and only the types $\alpha_{1},\alpha_{2},\beta_{1},\beta_{2},\delta_{1},\delta_{2},\varepsilon$ remain admissible. ∎

Example 4.3**.**

Concerning numerical realizations of Theorem 4.6, we refer to Corollary 4.1 for the parametrized infinite sequences $\mathbf{[7]}$ and $\mathbf{[2]}$ which realize item (1) and (4). (See also Tables 43 and 39 for the types $\vartheta$ and $\varepsilon$.) In all the other cases, there occurs type splitting:

The similarity class $\mathbf{[149]}$ partially realizes item (2). (See Table 38 for the type $\delta_{2}$.) Outside the range of our systematic investigations, we found that the similarity class $\mathbf{[1049]}$ realizes type $\zeta_{2}$. Realizations of the type $\vartheta$ are unknown up to now.

The similarity classes $\mathbf{[401]}$, $\mathbf{[151]}$ and $\mathbf{[101]}$ partially realize item (3). (See Tables 31, 32 and 40 for the types $\alpha_{1}$, $\alpha_{2}$ and $\zeta_{1}$.) Outside the range of our systematic investigations, we found that the similarity class $\mathbf{[1151]}$, resp. $\mathbf{[3251]}$, realizes type $\delta_{1}$, resp. $\delta_{2}$. Realizations of the types $\zeta_{2}$ and $\vartheta$ are unknown up to now.

The similarity classes $\mathbf{[139]}$, $\mathbf{[19]}$ and $\mathbf{[379]}$ completely realize item (5). (See Tables 35, 38 and 39 for the types $\beta_{2}$, $\delta_{2}$ and $\varepsilon$.)

The similarity classes $\mathbf{[31]}$, $\mathbf{[11]}$, $\mathbf{[191]}$ and $\mathbf{[211]}$ partially realize item (6). (See Tables 31, 32, 34 and 37 for the types $\alpha_{1}$, $\alpha_{2}$, $\beta_{1}$ and $\delta_{1}$.) Realizations of the types $\beta_{2}$, $\delta_{2}$ and $\varepsilon$ are unknown up to now.

4.4. Non-elementary $5$-class groups

Although most of the $5$-class groups of pure metacyclic fields $N$, maximal real subfields $M$ and pure quintic subfields $L$ are elementary abelian, there occur sparse examples with non-elementary structure. For instance, we have only $8$ occurrences within the range $2\leq D<10^{3}$ of our computations:

(1) $\mathrm{Cl}_{5}(N)\simeq C_{25}\times C_{5}^{3}$, $(V_{L},V_{M},V_{N};E)=(1,2,5*;6)$ for $D=259=7\cdot 37$ (type $\gamma$),

(2) $\mathrm{Cl}_{5}(N)\simeq C_{25}\times C_{5}^{7}$, $\mathrm{Cl}_{5}(M)\simeq C_{25}\times C_{5}^{3}$, $(V_{L},V_{M},V_{N};E)=(3,5*,9*;2)$ for $D=281$ prime

(type $\alpha_{1}$),

(3) $\mathrm{Cl}_{5}(N)\simeq C_{25}\times C_{5}^{5}$, $(V_{L},V_{M},V_{N};E)=(2,3,7*;4)$ for $D=465=3\cdot 5\cdot 31$ (type $\beta_{2}$),

(4) $\mathrm{Cl}_{5}(N)\simeq C_{25}\times C_{5}^{5}$, $(V_{L},V_{M},V_{N};E)=(2,3,7*;4)$ for $D=473=11\cdot 43$ (type $\beta_{2}$),

(5) $\mathrm{Cl}_{5}(N)\simeq C_{25}\times C_{5}^{6}$, $\mathrm{Cl}_{5}(M)\simeq C_{25}\times C_{5}^{2}$, $\mathrm{Cl}_{5}(L)\simeq C_{25}$, $(V_{L},V_{M},V_{N};E)=(2*,4*,8*;5)$

for $D=502=2\cdot 251$ (type $\beta_{1}$),

(6) $\mathrm{Cl}_{5}(N)\simeq C_{25}\times C_{5}^{5}$, $(V_{L},V_{M},V_{N};E)=(2,3,7*;4)$ for $D=590=2\cdot 5\cdot 59$ (type $\beta_{2}$),

(7) $\mathrm{Cl}_{5}(N)\simeq C_{25}^{2}\times C_{5}^{4}$, $\mathrm{Cl}_{5}(M)\simeq C_{25}\times C_{5}^{2}$, $(V_{L},V_{M},V_{N};E)=(2,4*,8*;5)$ for $D=620=2^{2}\cdot 5\cdot 31$

(type $\beta_{1}$),

(8) $\mathrm{Cl}_{5}(N)\simeq C_{25}\times C_{5}^{4}$, $\mathrm{Cl}_{5}(M)\simeq C_{25}\times C_{5}$, $\mathrm{Cl}_{5}(L)\simeq C_{25}$, $(V_{L},V_{M},V_{N};E)=(2*,3*,6*;3)$

for $D=955=5\cdot 191$ (type $\alpha_{2}$).

However, outside the range of systematic computations, we additionally found:

(a) $\mathrm{Cl}_{5}(N)\simeq C_{25}^{3}\times C_{5}$, $\mathrm{Cl}_{5}(M)\simeq C_{25}\times C_{5}$, $\mathrm{Cl}_{5}(L)\simeq C_{25}$, $(V_{L},V_{M},V_{N};E)=(2*,3*,7*;4)$

for $D=1049$ prime (type $\zeta_{2}$),

(b) $\mathrm{Cl}_{5}(N)\simeq C_{25}^{2}\times C_{5}^{6}$, $\mathrm{Cl}_{5}(M)\simeq C_{25}\times C_{5}^{3}$, $(V_{L},V_{M},V_{N};E)=(3,5*,10*;3)$ for $D=3001$ prime

(type $\alpha_{2}$),

(c) $\mathrm{Cl}_{5}(N)\simeq C_{25}^{5}\times C_{5}^{4}$, $\mathrm{Cl}_{5}(M)\simeq C_{25}^{2}\times C_{5}^{3}$, $\mathrm{Cl}_{5}(L)\simeq C_{25}\times C_{5}^{2}$, $(V_{L},V_{M},V_{N};E)=(4*,7*,14*;3)$

for $D=3251$ prime (type $\delta_{2}$),

(d) $\mathrm{Cl}_{5}(N)\simeq C_{25}^{2}\times C_{5}^{2}$, $\mathrm{Cl}_{5}(M)\simeq C_{25}\times C_{5}$, $\mathrm{Cl}_{5}(L)\simeq C_{25}$, $(V_{L},V_{M},V_{N};E)=(2*,3*,6*;3)$

for $D=5849$ prime (type $\delta_{2}$).

We point out that in all of the last four examples, the normal field $N$ is a Polya field, since the radicands $D$ are primes $\ell\equiv\pm 1\,(\mathrm{mod}\,25)$, the conductors are primes $f=\ell$, and thus all primitive ambiguous ideals are principal, generated by the radical $\delta=\sqrt[5]{D}$ and its powers. Consequently, there seems to be no upper bound for the complexity of $5$-class groups $\mathrm{Cl}_{5}(N)$ of pure metacyclic Polya fields $N$ in Theorem 4.4.

4.5. Refinement of DPF types by similarity classes

Based on the definition of similarity classes and prototypes in § 4.2, on the explicit listing of all prototypes in the range between $2$ and $10^{3}$ in the Tables 27 — 30, and on theoretical foundations in § 4.3, we are now in the position to establish the intended refinement of our $13$ differential principal factorization types into similarity classes in the Tables 31 — 43, as far as the range of our computations for normalized radicands $2\leq D<10^{3}$ is concerned. The cardinalities $\mathbf{\lvert M\rvert}$ refine the statistical evaluation in Table 26.

DPF types are characterized by the multiplet $(U,\eta,\zeta;A,I,R)$, refined Dedekind species, S, by the multiplet $(e_{0};t,u,v,m;n,s_{2},s_{4})$, and $5$-class groups by the multiplet $(V_{L},V_{M},V_{N};E)$.

DPF type $\alpha_{1}$ splits into $3$ similarity classes in the ground state $(V_{L},V_{M},V_{N})=(2,3,5)$ and $2$ similarity classes in the first excited state $(V_{L},V_{M},V_{N})=(3,5,9)$. Summing up the partial frequencies $6+3$ of these states in Table 31 yields the modest absolute frequency $9$ of type $\alpha_{1}$ in the range $2\leq D<10^{3}$, as given in Table 26. The logarithmic subfield unit index of type $\alpha_{1}$ is restricted to the single value $E=2$. Type $\alpha_{1}$ is the unique type with $2$-dimensional relative principal factorization, $R=2$.

The logarithmic subfield unit index of DPF type $\alpha_{2}$ can take two values, either $E=3$ or $E=1$. Type $\alpha_{2}$ with $E=3$ splits into $5$ similarity classes in the ground state $(V_{L},V_{M},V_{N})=(1,1,2)$ and $6$ similarity classes in the first excited state $(V_{L},V_{M},V_{N})=(2,3,6)$. Type $\alpha_{2}$ with $E=1$ splits into $7$ similarity classes in the ground state $(V_{L},V_{M},V_{N})=(2,2,4)$ and $4$ similarity classes in the first excited state $(V_{L},V_{M},V_{N})=(3,4,8)$. Summing up the partial frequencies $40+7$, resp. $24+4$, of these states in Table 32 yields the considerable absolute frequency $75$ of type $\alpha_{2}$ in the range $2\leq D<10^{3}$, as given in Table 26. Type $\alpha_{2}$ is the unique type with mixed intermediate and relative principal factorization, $I=R=1$.

DPF type $\alpha_{3}$ splits into $4$ similarity classes in the ground state $(V_{L},V_{M},V_{N})=(2,2,5)$ and $1$ similarity class in the first excited state $(V_{L},V_{M},V_{N})=(3,4,9)$. Summing up the partial frequencies $7+1$ of these states in Table 33 yields the modest absolute frequency $8$ of type $\alpha_{3}$ in the range $2\leq D<10^{3}$, as given in Table 26. The logarithmic subfield unit index of type $\alpha_{3}$ is restricted to the unique value $E=2$. Type $\alpha_{3}$ is the unique type with $2$-dimensional intermediate principal factorization, $I=2$.

The logarithmic subfield unit index of DPF type $\beta_{1}$ can take two values, either $E=3$ or $E=5$. Type $\beta_{1}$ with $E=3$ consists of $3$ similarity classes in the ground state $(V_{L},V_{M},V_{N})=(2,3,6)$. Type $\beta_{1}$ with $E=5$ splits into $5$ similarity classes in the ground state $(V_{L},V_{M},V_{N})=(1,2,4)$ and $2$ similarity classes in the first excited state $(V_{L},V_{M},V_{N})=(2,4,8)$. Summing up the partial frequencies $9$, resp. $12+2$, of these states in Table 34 yields the modest absolute frequency $23$ of type $\beta_{1}$ in the range $2\leq D<10^{3}$, as given in Table 26. Type $\beta_{1}$ is the unique type with mixed absolute and relative principal factorization, $A=2$ and $R=1$.

DPF type $\beta_{2}$ splits into $16$ similarity classes in the ground state $(V_{L},V_{M},V_{N})=(1,1,3)$ and $9$ similarity classes in the first excited state $(V_{L},V_{M},V_{N})=(2,3,7)$. Summing up the partial frequencies $146+15$ of these states in Table 35 yields the high absolute frequency $161$ of type $\beta_{2}$ in the range $2\leq D<10^{3}$, as given in Table 26. The logarithmic subfield unit index of type $\beta_{2}$ is restricted to the unique value $E=4$. Type $\beta_{2}$ is the unique type with mixed absolute and intermediate principal factorization, $A=2$ and $I=1$.

DPF type $\gamma$ splits into $6$ similarity classes in the ground state $(V_{L},V_{M},V_{N})=(0,0,1)$, $16$ similarity classes in the first excited state $(V_{L},V_{M},V_{N})=(1,2,5)$, and $5$ similarity classes in the second excited state $(V_{L},V_{M},V_{N})=(2,4,9)$. Summing up the partial frequencies $188+128+8$ of these states in Table 36 yields the maximal absolute frequency $324$ of type $\gamma$ among all $13$ types in the range $2\leq D<10^{3}$, as given in Table 26. The logarithmic subfield unit index of type $\gamma$ is restricted to the unique value $E=6$. Type $\gamma$ is the unique type with $3$-dimensional absolute principal factorization, $A=3$. However, the $1$-dimensional subspace $\Delta$ is formed by radicals, and only the complementary $2$-dimensional subspace is non-trivial.

The logarithmic subfield unit index of DPF type $\delta_{1}$ can take two values, either $E=4$ or $E=2$. Type $\delta_{1}$ with $E=4$ splits into $2$ similarity classes in the ground state $(V_{L},V_{M},V_{N})=(1,2,3)$. Type $\delta_{1}$ with $E=2$ consists of $1$ similarity class in the ground state $(V_{L},V_{M},V_{N})=(3,5,9)$. Summing up the partial frequencies $6+1$ of these states in Table 37 yields the modest absolute frequency $7$ of type $\delta_{1}$ in the range $2\leq D<10^{3}$, as given in Table 26. Type $\delta_{1}$ is a type with $1$-dimensional relative principal factorization, $R=1$.

DPF type $\delta_{2}$ splits into $4$ similarity classes in the ground state $(V_{L},V_{M},V_{N})=(1,1,2)$ and $1$ similarity class in the first excited state $(V_{L},V_{M},V_{N})=(2,3,6)$. Summing up the partial frequencies $52+1$ of these states in Table 38 yields the considerable absolute frequency $53$ of type $\delta_{2}$ in the range $2\leq D<10^{3}$, as given in Table 26. The logarithmic subfield unit index of type $\delta_{2}$ is restricted to the unique value $E=3$. Type $\delta_{2}$ is a type with $1$-dimensional intermediate principal factorization, $I=1$.

DPF type $\varepsilon$ splits into $3$ similarity classes in the ground state $(V_{L},V_{M},V_{N})=(0,0,0)$, $16$ similarity classes in the first excited state $(V_{L},V_{M},V_{N})=(1,2,4)$, and $3$ similarity classes in the second excited state $(V_{L},V_{M},V_{N})=(2,4,8)$. Summing up the partial frequencies $139+61+8$ of these states in Table 39 yields the high absolute frequency $208$ of type $\varepsilon$ in the range $2\leq D<10^{3}$, as given in Table 26. The logarithmic subfield unit index of type $\varepsilon$ is restricted to the unique value $E=5$. Type $\varepsilon$ is a type with $2$-dimensional absolute principal factorization, $A=2$.

The logarithmic subfield unit index of DPF type $\zeta_{1}$ is restricted to the unique value $E=5$. Type $\zeta_{1}$ consists of $1$ similarity class in the ground state $(V_{L},V_{M},V_{N})=(1,2,4)$. The frequency $1$ of this state in Table 40 coincides with the negligible absolute frequency $1$ of type $\zeta_{1}$ in the range $2\leq D<10^{3}$, as given in Table 26. Type $\zeta_{1}$ is a type with $1$-dimensional relative principal factorization, $R=1$.

DPF type $\zeta_{2}$ consists of $3$ similarity classes. The modest absolute frequency $5$ of type $\zeta_{2}$ in the range $2\leq D<10^{3}$, given in Table 26, is the sum $2+1+2$ of partial frequencies in Table 41. Type $\zeta_{2}$ only occurs with logarithmic subfield unit index $E=4$. It is a type with $1$-dimensional intermediate principal factorization, $I=1$.

DPF type $\eta$ splits in $2$ similarity classes, $[\mathbf{35}]$ and $[\mathbf{301}]$. The modest absolute frequency $7$ of type $\eta$ in the range $2\leq D<10^{3}$, given in Table 26, is the sum $6+1$ of partial frequencies in Table 42. Type $\eta$ only occurs with logarithmic subfield unit index $E=6$. It is a type with $2$-dimensional absolute principal factorization, $A=2$. However, it should be pointed out that outside of the range of our systematic investigations we found an excited state $(V_{L},V_{M},V_{N})=(1,2,5)$ for the similarity class $\mathbf{[1505]}$, where $1505=5\cdot 7\cdot 43$ has three prime divisors, additionally to the ground state $(V_{L},V_{M},V_{N})=(0,0,1)$.

DPF type $\vartheta$ splits into the unique finite similarity class $[\mathbf{5}]$ with only a single element and the infinite parametrized sequence $[\mathbf{7}]$ consisting of all prime radicands $D=q$ congruent to $\pm 7\,(\mathrm{mod}\,25)$. The small absolute frequency $19$ of type $\vartheta$ in the range $2\leq D<10^{3}$, given in Table 26, is the sum $\mathbf{\lvert 5\rvert}+\mathbf{\lvert 7\rvert}=1+18$ in Table 43. Since no theoretical argument disables the occurrence of type $\vartheta$ for composite radicands $D$ with prime factors $5$ and $q\equiv\pm 7\,(\mathrm{mod}\,25)$, we conjecture that such cases will appear in bigger ranges with $D>10^{3}$. Type $\vartheta$ only occurs with logarithmic subfield unit index $E=5$, and is the unique type where every unit of $K$ occurs as norm of a unit of $N$, that is $U=0$.

4.6. Increasing dominance of DPF type $\gamma$ for $T\to\infty$

In this final section, we want to show that the careful book keeping of similarity classes with representative prototypes in the Tables 31 – 43 is useful for the quantitative illumination of many other phenomena. For an explanation, we select the phenomenon of absolute principal factorizations.

The statistical distribution of DPF types in Table 26 has proved that type $\gamma$ with $324$ occurrences, that is $36\,\%$, among all $900$ fields $N=\mathbb{Q}(\zeta_{5},\sqrt[5]{D})$ with normalized radicands in the range $2\leq D<10^{3}$ is doubtlessly the high champion of all DPF types. This means that there is a clear trend towards the maximal possible extent of $3$-dimensional spaces of absolute principal factorizations, $A=3$, in spite of the disadvantage that the estimate $1\leq A\leq\min(3,T)$ in the formulas (4.3) and (4.4) of [15, Cor. 4.1] prohibits type $\gamma$ for conductors $f$ with $T\leq 2$ prime divisors.

For the following investigation, we have to recall that the number $T$ of all prime factors of $f^{4}=5^{e_{0}}\cdot q_{1}^{4}\ldots q_{t}^{4}$ is given by $T=t+1$ for fields of Dedekind’s species $1$, where $e_{0}\in\{2,6\}$, and by $T=t$ for fields of Dedekind’s species $2$, where $e_{0}=0$.

Conductors $f$ with $T=4$ prime factors occur in six tables,

$1$ case of type $\alpha_{2}$ in a single similarity class of Table 32,

$1$ case of type $\alpha_{3}$ in a single similarity class of Table 33,

$7$ cases of type $\beta_{1}$ in a single similarity class of Table 34,

$10$ cases of type $\beta_{2}$ in $5$ similarity classes of Table 35,

$126$ cases of type $\gamma$ in $16$ similarity classes of Table 36,

$8$ cases of type $\varepsilon$ in $3$ similarity classes of Table 39,

that is, a total of $153$ cases, with respect to the complete range $2\leq D<10^{3}$ of our computations. Consequently, we have an increase of type $\gamma$ from $36.0\,\%$, with respect to the entire database, to $\frac{126}{153}=82.4\,\%$, with respect to $T=4$.

The feature is even aggravated for conductors $f$ with $T=5$ prime factors, which exclusively occur in Table 36. There are $4$ similarity classes with $T=5$, namely $\mathbf{[462]}$, $\mathbf{[546]}$, $\mathbf{[798]}$, $\mathbf{[858]}$, with a total of $6$ elements, all $(100\,\%)$ with associated fields of type $\gamma$.

5. Acknowledgements

We gratefully acknowledge that our research was supported by the Austrian Science Fund (FWF): projects J 0497-PHY and P 26008-N25. This work is dedicated to the memory of Charles J. Parry ($\dagger$ 25 December 2010) who suggested a numerical investigation of pure quintic number fields.