{"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/good-elliptic-curves-with-a-specified-torsion","title":"Good elliptic curves with a specified torsion subgroup","arxiv_id":"2012.12475","date":"2020-12-23","proceeding":null,"authors":["Alexander J. Barrios"],"abstract":"An elliptic curve $E$ over $\\mathbb{Q}$ is said to be good if $N_{E}^{6}<\\max\\!\\left\\{ \\left\\vert c_{4}^{3}\\right\\vert ,c_{6}^{2}\\right\\} $ where $N_{E}$ is the conductor of $E$ and $c_{4}$ and $c_{6}$ are the invariants associated to a global minimal model of $E$. In this article, we generalize Masser's Theorem on the existence of infinitely many good elliptic curves with full $2$-torsion. Specifically, we prove via constructive methods that for each of the fifteen torsion subgroups $T$ allowed by Mazur's Torsion Theorem, there are infinitely many good elliptic curves $E$ with $E\\!\\left(\\mathbb{Q}\\right) _{\\text{tors}}\\cong T$.","url_abs":"https://arxiv.org/abs/2012.12475v2","url_pdf":"https://arxiv.org/pdf/2012.12475v2.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":"good-elliptic-curves-with-a-specified-torsion","repo_url":"https://github.com/alexanderbarrios/code_for_good_ec","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}