{"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/explicit-classification-of-isogeny-graphs-of","title":"Explicit classification of isogeny graphs of rational elliptic curves","arxiv_id":"2208.05603","date":"2022-08-11","proceeding":null,"authors":["Alexander J. Barrios"],"abstract":"Let $n>1$ be an integer such that $X_{0}\\!\\left( n\\right) $ has genus $0$, and let $K$ be a field of characteristic $0$ or relatively prime to $6n$. In this article, we explicitly classify the isogeny graphs of all rational elliptic curves that admit a non-trivial isogeny over $\\mathbb{Q}$. We achieve this by introducing $56$ parameterized families of elliptic curves $\\mathcal{C}_{n,i}(t,d)$ defined over $K(t,d)$, which have the following two properties for a fixed $n$: the elliptic curves $\\mathcal{C}_{n,i}(t,d)$ are isogenous over $K(t,d)$, and there are integers $k_{1}$ and $k_{2}$ such that the $j$-invariants of $\\mathcal{C}_{n,k_{1}}(t,d)$ and $\\mathcal{C}_{n,k_{2}}(t,d)$ are given by the Fricke parameterizations. As a consequence, we show that if $E$ is an elliptic curve over a number field $K$ with isogeny class degree divisible by $n\\in\\left\\{4,6,9\\right\\} $, then there is a quadratic twist of $E$ that is semistable at all primes $\\mathfrak{p}$ of $K$ such that $\\mathfrak{p}\\nmid n$.","url_abs":"https://arxiv.org/abs/2208.05603v1","url_pdf":"https://arxiv.org/pdf/2208.05603v1.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":"explicit-classification-of-isogeny-graphs-of","repo_url":"https://github.com/alexanderbarrios/explicit_isogenies","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"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}