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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1102.4291","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.NT","submitted_at":"2011-02-21T17:54:33Z","cross_cats_sorted":[],"title_canon_sha256":"2c1ab0fa2469e12adb8cd76d131a482d193680c34f5146374213e6a8a83b8c1b","abstract_canon_sha256":"5de5834237574973e69d7bab5943b7d1fa2a5dd7d54c754824a410f6b28f0f9d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:31:27.796722Z","signature_b64":"gj8lzS10Nf34CnGSRSaCF2nsgMUdBzm/ktpvJTcbzzP0JFiarJP4JRhLTXHTZpWbYxfF80P1F3XocUFRbNHZCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f67c2e144654901a4f73551b6cd460bf9644fa67d2f34ef68a1a102ccd34d8b3","last_reissued_at":"2026-05-18T02:31:27.796278Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:31:27.796278Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\theta=2\\pi/3$. 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