{"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/minimal-models-of-rational-elliptic-curves","title":"Minimal models of rational elliptic curves with non-trivial torsion","arxiv_id":"2001.01016","date":"2020-01-04","proceeding":null,"authors":["Alexander J. Barrios"],"abstract":"In this paper, we explicitly classify the minimal discriminants of all elliptic curves $E/\\mathbb{Q}$ with a non-trivial torsion subgroup. This is done by considering various parameterized families of elliptic curves with the property that they parameterize all elliptic curves $E/\\mathbb{Q}$ with a non-trivial torsion point. We follow this by giving admissible change of variables, which give a global minimal model for $E$. We also provide necessary and sufficient conditions on the parameters of these families to determine the primes at which $E$ has additive reduction. In addition, we use these parameterized families to give new proofs of results due to Frey and Flexor-Oesterl\\'{e} pertaining to the primes at which an elliptic curve over a number field $K$ with a non-trivial $K$-torsion point can have additive reduction.","url_abs":"https://arxiv.org/abs/2001.01016v3","url_pdf":"https://arxiv.org/pdf/2001.01016v3.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":"minimal-models-of-rational-elliptic-curves","repo_url":"https://github.com/alexanderbarrios/code_for_rational_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}