Cyclic cubic number fields with harmonically balanced capitulation

TL;DR

This paper proves that 689347 is the smallest conductor for a cyclic cubic number field with a maximal unramified pro-3-extension having a specific automorphism group structure and harmonic transfer kernel balance.

Contribution

It establishes the minimal conductor for such cyclic cubic fields with complex automorphism group properties and harmonic transfer kernel balance.

Findings

Smallest conductor identified as 689347.

Automorphism group of order 6561 with specific properties.

Harmonically balanced transfer kernels in S(13).

Abstract

It is proved that c = 689347 = 31*37*601 is the smallest conductor of a cyclic cubic number field K whose maximal unramified pro-3-extension E = F(3,infinity,K) possesses an automorphism group G = Gal(E/K) of order 6561 with coinciding relation and generator rank d2(G) = d1(G) = 3 and harmonically balanced transfer kernels kappa(G) in S(13).