Simplest cubic fields with small class number

Akinari Hoshi; Hiroaki Iida

Simplest cubic fields with small class number

Number Theory 2026-04-07 v2

Authors: Akinari Hoshi , Hiroaki Iida

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.

Keywords

class number of quadratic fields

Cite

@article{arxiv.2603.18802,
  title  = {Simplest cubic fields with small class number},
  author = {Akinari Hoshi and Hiroaki Iida},
  journal= {arXiv preprint arXiv:2603.18802},
  year   = {2026}
}

Comments

23 pages. We would like to thank Stephane Louboutin for helpful explanations concerning Theorem 1.8 which improved the proof of Theorem 1.13. For Theorem 1.14, the result is also improved as integers m up to 10^7 instead of 4x10^6. Only one additional case was found as in Table 3

Simplest cubic fields with small class number

Abstract

Keywords

Cite

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