Simplest cubic fields with small class number

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 $f_m(X)=X^3-m{X}^2-(m+3)X-1$. Let ${\mathcal{O}}_{L_m}$ be the ring of integers of $L_m$. By using PARI/GP, we determine 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 $m\ge -1$ such that $h_m\le 1000$. We also show that if $-1\le m\le 10^7$, 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$$)$ integers $m$ with $-1\le m\le 10^7$ such that $h_m=1$ $($resp. $3$, $4$, $7$, $9$, $12$, $13$$)$ which are given explicitly. All computations of the class numbers and class groups of the listed fields are certified unconditionally. Under the GRH $($Generalized Riemann Hypothesis$)$, we confirm that these are the only such integers $m$ in the range $-1\le m\le 10^7$. We conjecture that the $26$ integers $m$ obtained above with $h_m=1$ are the only integers $m\ge -1$ with $h_m=1$.

23 pages, Abstract modified, Remark 1.15, Conjecture 1.16 and Remark 1.17 added