{"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/counting-5-isogenies-of-elliptic-curves-over","title":"Counting 5-isogenies of elliptic curves over $\\mathbb{Q}$","arxiv_id":"2504.07750","date":"2025-04-10","proceeding":null,"authors":["Santiago Arango-Piñeros","Changho Han","Oana Padurariu","Sun Woo Park"],"abstract":"We show that the number of $5$-isogenies of elliptic curves defined over $\\mathbb{Q}$ with naive height bounded by $H > 0$ is asymptotic to $C_5\\cdot H^{1/6} (\\log H)^2$ for some explicitly computable constant $C_5 > 0$. This settles the asymptotic count of rational points on the genus zero modular curves $X_0(m)$. We leverage an explicit $\\mathbb{Q}$-isomorphism between the stack $\\mathscr{X}_0(5)$ and the generalized Fermat equation $x^2 + y^2 = z^4$ with $\\mathbb{G}_m$-action of weights $(4, 4, 2)$.","url_abs":"https://arxiv.org/abs/2504.07750v2","url_pdf":"https://arxiv.org/pdf/2504.07750v2.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":"counting-5-isogenies-of-elliptic-curves-over","repo_url":"https://github.com/sarangop1728/counting-5-isogenies","is_official":1,"mentioned_in_paper":0,"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}