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Torsion of elliptic curves over number fields of degree 1010

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Filip Najman ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

We determine all the groups that appear as the torsion group of an elliptic curve over a number field of degree 1010. There are 6969 such groups. For 44 of them, namely Z/2Z×Z/26Z\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/26\mathbb{Z}, Z/3Z×Z/15Z\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/15\mathbb{Z}, Z/3Z×Z/18Z\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/18\mathbb{Z} and Z/6Z×Z/12Z\mathbb{Z}/6\mathbb{Z}\times\mathbb{Z}/12\mathbb{Z}, we also determine all the elliptic curves over number fields of degree 1010 that realize them, and there are only finitely many. These four groups were not known to occur over number fields of degree 1010. The proofs combine the global method of Derickx, Hecke sieves, formal immersion criteria at the prime 22, and Riemann-Roch computations on modular curves and their quotients over finite fields.

Author comments: Code: https://github.com/F-Najman/deg10_torsion (large data files in its release large-files-2026-11).

Provenance statement

The mathematical content of this paper was produced by AI models. OpenAI's GPT-6.1 Sol produced almost all of the mathematics of the first version of the proof and carried out most of the computations. Anthropic's Claude Opus 5.5 reviewed that version, independently recomputed some of the computations and all the explicit examples, simplified several of the arguments, wrote the manuscript and organized the code repository. I posed the question, guided the work and made the editorial decisions. I have not yet fully checked the proofs; mathematically, the paper should be regarded as raw data.

Tools used

OpenAI
ChatGPTVersion 6.1 Sol
Anthropic
Claude OpusVersion 5.5
Magma
MagmaVersion 2.30-1
SageMath
SageMathVersion 10.7

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Version history

  1. v1 · 124 pagesSubmitted by Filip NajmanCode: https://github.com/F-Najman/deg10_torsion (large data files in its release large-files-2026-11).CurrentNov 09, 2026