{"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/cyclic-isogenies-of-elliptic-curves-over-a","title":"Cyclic isogenies of elliptic curves over fixed quadratic fields","arxiv_id":"2206.08891","date":"2022-06-17","proceeding":null,"authors":["Barinder S. Banwait","Filip Najman","Oana Padurariu"],"abstract":"Building on Mazur's 1978 work on prime degree isogenies, Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over $\\mathbb{Q}$. Although more than 40 years have passed, the determination of cyclic isogenies of elliptic curves over a single other number field has hitherto not been realised. In this paper we develop a procedure to assist in establishing such a determination for a given quadratic field. Executing this procedure on all quadratic fields $\\mathbb{Q}(\\sqrt{d})$ with $|d| < 10^4$ we obtain, conditional on the Generalised Riemann Hypothesis, the determination of cyclic isogenies of elliptic curves over $19$ quadratic fields, including $\\mathbb{Q}(\\sqrt{213})$ and $\\mathbb{Q}(\\sqrt{-2289})$. To make this procedure work, we determine all of the finitely many quadratic points on the modular curves $X_0(125)$ and $X_0(169)$, which may be of independent interest.","url_abs":"https://arxiv.org/abs/2206.08891v2","url_pdf":"https://arxiv.org/pdf/2206.08891v2.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":"cyclic-isogenies-of-elliptic-curves-over-a","repo_url":"https://github.com/barinderbanwait/quadratic_isogeny_composites","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"cyclic-isogenies-of-elliptic-curves-over-a","repo_url":"https://github.com/barinderbanwait/quadratic_kenku_solver","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}