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Simplest cubic fields with small class number

Akinari Hoshi, Hiroaki Iida

Abstract

Let $m\in\mathbb{Z}$ be an integer and $L_m=\mathbb{Q}(α)$ be the simplest cubic field with class number $h_m$ and conductor $\mathfrak{f}_m$ where $α$ 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 $[\mathcal{O}_{L_m}:\mathbb{Z}[α]]=1$ $($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 $-1\leq m\leq 4\cdot 10^6$, then $h_m<16$ holds for $137=26+31+10+10+36+21+3$ integers $m$. More precisely, there exist $26$ $($resp. $31$, $10$, $10$, $36$, $21$, $3$$)$ integers $m$ with $-1\leq m\leq 4\cdot 10^6$ such that $h_m=1$ $($resp. $3$, $4$, $7$, $9$, $12$, $13$$)$ which are given explicitly.

Simplest cubic fields with small class number

Abstract

Let be an integer and be the simplest cubic field with class number and conductor where is a root of . Let be the ring of integers of . By using PARI/GP, we confirm that if resp. , , i.e. resp. , , then there exist exactly (resp. , ) integers such that . We also show that if , then holds for integers . More precisely, there exist resp. , , , , , integers with such that resp. , , , , , which are given explicitly.
Paper Structure (4 sections, 12 theorems, 20 equations)

This paper contains 4 sections, 12 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thmain1']} and Theorem \ref{['thmain2']}
  3. PARI/GP computations: The simplest cubic fields $L_m=\mathbbm{Q}(\alpha)$ with class number $h_m=|\mathrm{Cl}(L_m)|\leq 1000$ and $[\mathcal{O}_{L_m}:\mathbbm{Z}[\alpha]]=1,3,27$
  4. PARI/GP computations: The simplest cubic fields $L_m$ with class number $h_m=|\mathrm{Cl}(L_m)|<16$ for $-1\leq m\leq 4\cdot 10^6$

Key Result

Theorem 1.1

Let $m\in\mathbbm{Z}$ be an integer and $L_m=\mathbbm{Q}(\alpha)$ be the simplest cubic field where $\alpha$ is a root of $f_m(X)=X^3-mX^2-(m+3)X-1$. Write $m^2+3m+9=bc^3$ for $b,c\geq 0$ with cubefree integer $b$. Then the conductor $\mathfrak{f}_m$ of $L_m$ is given by where In particular, $\mathfrak{f}_m=p$ is prime if and only if (i) $m\not\equiv 0\pmod{3}\ {\rm or}\ m\equiv 12\ ({\rm mod}\

Theorems & Definitions (13)

  • Theorem 1.1: Gras Gra73, Gra74, Gra86, Cusick Cus83 ($m^2+3m+9$ is squarefree), Washington Was87 $(m\not\equiv 3\,({\rm mod}\ 9))$, Komatsu Kom04, Kom07, Häberle Hab10
  • Corollary 1.2
  • Theorem 1.3: Shanks Sha74
  • Theorem 1.4: see e.g. Gras Gra75, see also Lemmermeyer Lem13 for the ambiguous class number formula $|\mathrm{Cl}(L_m)^{C_3}|=3^{r-1}$
  • Theorem 1.5: Lettl Let86
  • Theorem 1.6: Lettl Let86
  • Remark 1.7
  • Theorem 1.8: Hoshi Hos11
  • Theorem 1.9: Hoshi Hos11
  • Theorem 1.10: Okazaki Oka, Hoshi Hos11
  • ...and 3 more