Simplest cubic fields with small class number
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866913006587215872 |
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| author | Hoshi, Akinari Iida, Hiroaki |
| author_facet | Hoshi, Akinari Iida, Hiroaki |
| contents | 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 $-1\leq m\leq 10^7$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_18802 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Simplest cubic fields with small class number Hoshi, Akinari Iida, Hiroaki Number Theory 11D25, 11D59, 11R16, 11Y40, 12F10 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 $-1\leq m\leq 10^7$, 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. |
| title | Simplest cubic fields with small class number |
| topic | Number Theory 11D25, 11D59, 11R16, 11Y40, 12F10 |
| url | https://arxiv.org/abs/2603.18802 |