{"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/reduced-minimal-models-and-torsion","title":"Reduced minimal models and torsion","arxiv_id":"2301.09488","date":"2023-01-23","proceeding":null,"authors":["Alexander J. Barrios"],"abstract":"Let $E/\\mathbb{Q}$ be an elliptic curve. The reduced minimal model of $E$ is a global minimal model $y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}$ which satisfies the additional conditions that $a_{1},a_{3}\\in \\{0,1\\}$ and $a_{2}\\in\\{0,\\pm1\\}$. The reduced minimal model of $E$ is unique, and in this article, we explicitly classify the reduced minimal model of an elliptic curve $E/\\mathbb{Q}$ with a non-trivial torsion point. We obtain this classification by first showing that the reduced minimal model of $E$ is uniquely determined by a congruence on $c_6$ modulo $24$. We then apply this result to parameterized families of elliptic curves to deduce our main result. We also show that the reduction at $2$ and $3$ of $E$ affects the reduced minimal model of $E$.","url_abs":"https://arxiv.org/abs/2301.09488v1","url_pdf":"https://arxiv.org/pdf/2301.09488v1.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":"reduced-minimal-models-and-torsion","repo_url":"https://github.com/alexanderbarrios/reduced_minimal_models","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}