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Rational points on the non-split Cartan modular curve of level 121121

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

Contributed by Santiago Arango-Piñeros ↗, Filip Najman ↗, David Zureick-Brown ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

We show that the modular curve Xns+(121)X_{\mathrm{ns}}^+(121), attached to the normalizer of a non-split Cartan subgroup of GL2(Z/121Z)\mathrm{GL}_2(\mathbb{Z}/121\mathbb{Z}), has exactly 77 rational points, all of them CM points. Together with the work of Rouse, Sutherland and Zureick-Brown, this completes the classification of the possible 1111-adic images of Galois of elliptic curves over Q\mathbb{Q} without complex multiplication. We do not use a model of Xns+(121)X_{\mathrm{ns}}^+(121), which has genus 511511. Instead we use the map Xns+(121)→Xns+(11)X_{\mathrm{ns}}^+(121)\rightarrow X_{\mathrm{ns}}^+(11), whose target is an elliptic curve of rank 11, a descent with the cusps of Xns+(11)X_{\mathrm{ns}}^+(11) and a Mordell-Weil sieve to reduce the problem to 77 hyperelliptic curves of genus 55 over a number field of degree 1515. We deal with these curves by a version of Stoll's Selmer group Chabauty method for the isogeny 1−ζ111-\zeta_{11} of their Jacobians.

Author comments: 19 pages. Code, logs and certificates: https://github.com/F-Najman/Xns-121-

Provenance statement

The mathematical content of this paper was produced by AI models. Anthropic's Claude Opus 5.5, 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. Anthropic's Claude Opus 5.5 reviewed these proofs, checked the references, wrote the manuscript and organized the code repository. We have not yet fully checked the proofs.

Tools used

Anthropic
Claude OpusVersion 5.5
OpenAI
GPTVersion 6.1 Sol
OpenAI
GPTVersion 6 Astra
Magma
MagmaVersion V2.30-1

References

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Version history

  1. v1 · 19 pagesSubmitted by Filip Najman19 pages. Code, logs and certificates: https://github.com/F-Najman/Xns-121-CurrentNov 09, 2026