{"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/on-families-of-elliptic-curves-e-p-q-y-2-x-3","title":"On families of elliptic curves $E_{p,q}:y^2=x^3-pqx$ that intersect the same line $L_{a,b}:y=\\frac{a}{b}x$ of rational slope","arxiv_id":"2401.00215","date":"2023-12-30","proceeding":null,"authors":["Eldar Sultanow","Anja Jeschke","Amir Darwish Tfiha","Madjid Tehrani","William J Buchanan"],"abstract":"Let $p$ and $q$ be two distinct odd primes, $p<q$ and $E_{p,q}:y^2=x^3-pqx$ be an elliptic curve. Fix a line $L_{a.b}:y=\\frac{a}{b}x$ where $a\\in \\mathbb{Z},b\\in \\mathbb{N}$ and $(a,b)=1$. We study sufficient conditions that $p$ and $q$ must satisfy so that there are infinitely many elliptic curves $E_{p,q}$ that intersect $L_{a,b}$.","url_abs":"https://arxiv.org/abs/2401.00215v3","url_pdf":"https://arxiv.org/pdf/2401.00215v3.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":"on-families-of-elliptic-curves-e-p-q-y-2-x-3","repo_url":"https://github.com/sultanow/elliptic_curves","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"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}