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arXiv · 2605.14226

Kodaira-Neron statistics for rational elliptic curves with $j$-invariant 0 and 1728

John Cullinan·Sebastian Sargenti
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Abstract

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-N\'eron types at 3 and 2, respectively. We also give related statistics by holding the torsion subgoup and isogeny-torsion graph constant.

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John Cullinan, Sebastian Sargenti. 2026-05-14. Kodaira-Neron statistics for rational elliptic curves with $j$-invariant 0 and 1728. https://arxiv.org/abs/2605.14226

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