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Torsion of elliptic curves over number fields of degree 66, 77, 88 and 99

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

Contributed by Filip Najman ↗

Version 1 / Nov 04, 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 dd for d=6d=6, 77, 88 and 99. Apart from the groups that occur infinitely often, exactly 44 groups occur in degree 66, 33 in degree 77, 88 in degree 88 and 99 in degree 99, and among them the groups Z/2Z×Z/24Z\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/24\mathbb{Z} over sextic fields, Z/3Z×Z/15Z\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/15\mathbb{Z} over octic fields and Z/2Z×Z/26Z\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/26\mathbb{Z} over nonic fields were not known to occur. For 1313 of these 2424 pairs of a degree and a group we also determine all the elliptic curves that realize them. The proofs combine the global method of Derickx, Hecke sieves, formal immersion criteria at the prime 22 with Hecke operators, and Riemann-Roch computations on modular curves and their quotients over finite fields.

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 versions of the proofs, separately for degrees 6, 7, 8 and 9, and carried out most of the computations, starting in mid-October 2026. Anthropic's Claude Opus 5.5 reviewed these versions, independently recomputed some of the computations and the explicit examples, merged the proofs into one paper, wrote the manuscript and organized the code repositories. The merged paper was also reviewed by independent AI agents (Claude Opus 5.5, ChatGPT-6.1 Sol and ChatGPT-6 Astra). I posed the questions, guided the work and made the editorial decisions, and I worked to keep the presentation understandable and well organized. I have carefully checked, understood and rewritten the degree 6 part; that version will appear separately on arXiv. Mathematically, the present paper should be regarded as raw data: I have not yet fully checked the parts on degrees 7, 8 and 9, and its degree 6 part differs in places from the checked version.

Tools used

OpenAI
ChatGPTVersion 6.1 Sol
Anthropic
Claude OpusVersion 5.5
Magma
MagmaVersion c2.30-1

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Version history

  1. v1Submitted by Filip NajmanInitial depositCurrentNov 03, 2026