Simplest cubic fields with small class number

TL;DR

This paper investigates the class numbers of simplest cubic fields, confirming the number of fields with class number up to 1000 for specific conductors using PARI/GP, and identifying explicit examples with small class numbers.

Contribution

It provides a detailed computational analysis of simplest cubic fields' class numbers, including explicit examples and counts for fields with class number less than or equal to 1000.

Findings

Exactly 581, 80, and 142 fields with specific conductors have class number ≤ 1000.

For m between -1 and 10^7, 138 fields have class number less than 16.

Explicit examples of fields with class numbers 1, 3, 4, 7, 9, 12, 13 are given.

Abstract

Let $m\in\mathbb{Z}$ be an integer and $L_m=\mathbb{Q}(\alpha)$ be the simplest cubic field with class number $h_m$ and conductor $\mathfrak{f}_m$ where $\alpha$ is a root of $f_m(X)=X^3-mX^2-(m+3)X-1$. Let $\mathcal{O}_{L_m}$ be the ring of integers of $L_m$. By using PARI/GP, we confirm that if $[\mathcal{O}_{L_m}:\mathbb{Z}[\alpha]]=1$ $($resp. $3$, $27$$)$, i.e. $m^2+3m+9=\mathfrak{f}_m$ $($resp. $3\mathfrak{f}_m$, $27\mathfrak{f}_m$$)$, then there exist exactly $581$ (resp. $80$, $142$) integers $m\geq -1$ such that $h_m\leq 1000$. We also show that if $-1\leq m\leq 10^7$, then $h_m<16$ holds for $138=26+31+11+10+36+21+3$ integers $m$. More precisely, there exist $26$ $($resp. $31$, $11$, $10$, $36$, $21$, $3$$)$ integers $m$ with $-1\leq m\leq 10^7$ such that $h_m=1$ $($resp. $3$, $4$, $7$, $9$, $12$, $13$$)$ which are given explicitly.