Kodaira-Neron statistics for rational elliptic curves with $j$-invariant 0 and 1728
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914565103550464 |
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| author | Cullinan, John Sargenti, Sebastian |
| author_facet | Cullinan, John Sargenti, Sebastian |
| contents | Elliptic curves over $\Q$ with $j$-invariant 0 or 1728 have additive reduction at all primes of bad reduction. In addition, all elliptic curves with $j$-invariant 0 have bad reduction at $p=3$ and all elliptic curves with $j$-invariant 1728 have bad reduction at $p=2$. In this paper we count elliptic curves with $j$-invariant 0 and 1728 by height and determine asymptotics for the various Kodaira-Néron types at 3 and 2, respectively. We also give related statistics by holding the torsion subgoup and isogeny-torsion graph constant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_14226 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Kodaira-Neron statistics for rational elliptic curves with $j$-invariant 0 and 1728 Cullinan, John Sargenti, Sebastian Number Theory Elliptic curves over $\Q$ with $j$-invariant 0 or 1728 have additive reduction at all primes of bad reduction. In addition, all elliptic curves with $j$-invariant 0 have bad reduction at $p=3$ and all elliptic curves with $j$-invariant 1728 have bad reduction at $p=2$. In this paper we count elliptic curves with $j$-invariant 0 and 1728 by height and determine asymptotics for the various Kodaira-Néron types at 3 and 2, respectively. We also give related statistics by holding the torsion subgoup and isogeny-torsion graph constant. |
| title | Kodaira-Neron statistics for rational elliptic curves with $j$-invariant 0 and 1728 |
| topic | Number Theory |
| url | https://arxiv.org/abs/2605.14226 |