{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/computing-quadratic-points-on-modular-curves","title":"Computing quadratic points on modular curves $X_0(N)$","arxiv_id":"2303.12566","date":"2023-03-22","proceeding":null,"authors":["Nikola Adžaga","Timo Keller","Philippe Michaud-Jacobs","Filip Najman","Ekin Ozman","Borna Vukorepa"],"abstract":"In this paper we improve on existing methods to compute quadratic points on modular curves and apply them to successfully find all the quadratic points on all modular curves $X_0(N)$ of genus up to $8$, and genus up to $10$ with $N$ prime, for which they were previously unknown. The values of $N$ we consider are contained in the set \\[ \\mathcal{L}=\\{58, 68, 74, 76, 80, 85, 97, 98, 100, 103, 107, 109, 113, 121, 127 \\}.\\] We obtain that all the non-cuspidal quadratic points on $X_0(N)$ for $N\\in \\mathcal{L}$ are CM points, except for one pair of Galois conjugate points on $X_0(103)$ defined over $\\mathbb{Q}(\\sqrt{2885})$. We also compute the $j$-invariants of the elliptic curves parametrised by these points, and for the CM points determine their geometric endomorphism rings.","url_abs":"https://arxiv.org/abs/2303.12566v2","url_pdf":"https://arxiv.org/pdf/2303.12566v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"computing-quadratic-points-on-modular-curves","repo_url":"https://github.com/timokellermath/quadraticpoints","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}