{"paper":{"title":"On the high rank $\\pi/3$ and $2\\pi/3$-congruent number elliptic curves","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ali S. Janfada, Andrej Dujella, Juan C. Peral, Sajad Salami","submitted_at":"2011-02-21T17:54:33Z","abstract_excerpt":"Consider the elliptic curves given by $ E_{n,\\theta}:\\quad y^2=x^3+2s n x^2-(r^2-s^2) n^2 x $ where $0 < \\theta< \\pi$, $\\cos(\\theta)=s/r$ is rational with $0\\leq |s| <r$ and $\\gcd (r,s)=1$. These elliptic curves are related to the $\\theta$-congruent number problem as a generalization of the congruent number problem. For fixed $\\theta$ this family corresponds to the quadratic twist by $n$ of the curve $E_{\\theta}: \\,\\, y^2=x^3+2s x^2-(r^2-s^2) x.$ We study two special cases $\\theta=\\pi/3$ and $\\theta=2\\pi/3$. We have found a subfamily of $n=n(w)$ having rank at least $3$ over ${\\mathbb Q}(w)$ a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.4291","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}