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Points of prime order on elliptic curves over number fields of degree 99 to 2222

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

Contributed by Filip Najman ↗

Version 1 / Nov 05, 2026 / CC BY 4.0

Abstract

Let S(d)S(d) be the set of primes pp for which there exists an elliptic curve over a number field of degree dd with a point of order pp. The sets S(d)S(d) are known for d≤8d\leq 8. We determine S(d)S(d) for all 9≤d≤229\leq d\leq 22. For example, the largest prime in S(22)S(22) is 6767, and S(16)S(16) consists of the primes up to 3737. To reduce the problem to finitely many primes we prove a version of Oesterlé's bound which uses reduction modulo 22 instead of modulo 33, and which is much smaller for d≥13d\geq 13. The remaining primes are dealt with using results of Derickx and Stoll and extensive computations with the modular curves X1(p)X_1(p) over F2\mathbb{F}_2 for p≤139p\leq 139.

Author comments: Code: https://github.com/F-Najman/Sd. Computational certificates: https://doi.org/10.7910/DVN/5HHEY4.

Provenance statement

The mathematical content of this paper was produced by AI models. OpenAI's GPT-6.1 Sol and GPT-6 Astra produced almost all of the mathematics of the first versions of the proofs and carried out the computations, between 28 October and 3 November 2026. Anthropic's Claude Opus 5.5 reviewed these proofs, without rerunning the computations, checked the references, wrote the manuscript and organized the code repository. The manuscript was also reviewed by an independent AI agent (Codex). For primes between 10^5 and 4·10^5, the large-prime step was checked independently against data of Michael Stoll. 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
OpenAI
ChatGPTVersion 6 Astra
Anthropic
Claude OpusVersion 5.5
Magma
MagmaVersion 2.30-1
SageMath
SageMathVersion 10.7

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Version history

  1. v1 · 31 pagesSubmitted by Filip NajmanCode: https://github.com/F-Najman/Sd. Computational certificates: https://doi.org/10.7910/DVN/5HHEY4.CurrentNov 05, 2026