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We express the number of rational points of $E$ over $\\mathbb{F}_p$ using the Gaussian hypergeometric series $\\displaystyle {_2F_1}\\left(\\begin{matrix}\n  \\phi&\\phi\n  {} & \\epsilon\n  \\end{matrix}\\Big| x\\right)$ where $\\epsilon$ and $\\phi$ are the trivial and quadratic characters over $\\mathbb{F}_p$ respect"},"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":"1501.03526","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-01-14T21:54:40Z","cross_cats_sorted":[],"title_canon_sha256":"76a0ba7915d3f7bba1dfc69390416b435d266a543f5bbac0dcf94afc5cf0da81","abstract_canon_sha256":"b9810f42adf84ed4717b74f31ae638c476e197cd7fd3a726a52ae32ffe0d3113"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:19:38.865521Z","signature_b64":"T6u0NBhDLsIkjH5awJleK8KAJJ99fz0FK0zqjKhKWhPqi8RpxqmG7pieuPXfh5qf0Q6wB15kLuCMDLAb+8xwCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"907f069172a1aa5011d29061bafe794a7bde0a7296d3cac25b4f68ef94b51198","last_reissued_at":"2026-05-18T01:19:38.864947Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:19:38.864947Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Edwards Curves and Gaussian Hypergeometric Series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Mohammad Sadek, Nermine El-Sissi","submitted_at":"2015-01-14T21:54:40Z","abstract_excerpt":"Let $E$ be an elliptic curve described by either an Edwards model or a twisted Edwards model over $\\mathbb{F}_p$, namely, $E$ is defined by one of the following equations $x^2+y^2=a^2(1+x^2y^2),\\, a^5-a\\not\\equiv 0$ mod $p$, or, $ax^2+y^2=1+dx^2y^2,\\,ad(a-d)\\not\\equiv0$ mod $p$, respectively. We express the number of rational points of $E$ over $\\mathbb{F}_p$ using the Gaussian hypergeometric series $\\displaystyle {_2F_1}\\left(\\begin{matrix}\n  \\phi&\\phi\n  {} & \\epsilon\n  \\end{matrix}\\Big| x\\right)$ where $\\epsilon$ and $\\phi$ are the trivial and quadratic characters over $\\mathbb{F}_p$ respect"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.03526","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1501.03526","created_at":"2026-05-18T01:19:38.865037+00:00"},{"alias_kind":"arxiv_version","alias_value":"1501.03526v1","created_at":"2026-05-18T01:19:38.865037+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.03526","created_at":"2026-05-18T01:19:38.865037+00:00"},{"alias_kind":"pith_short_12","alias_value":"SB7QNELSUGVF","created_at":"2026-05-18T12:29:39.896362+00:00"},{"alias_kind":"pith_short_16","alias_value":"SB7QNELSUGVFAEOS","created_at":"2026-05-18T12:29:39.896362+00:00"},{"alias_kind":"pith_short_8","alias_value":"SB7QNELS","created_at":"2026-05-18T12:29:39.896362+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/SB7QNELSUGVFAEOSSBQ3V7TZJJ","json":"https://pith.science/pith/SB7QNELSUGVFAEOSSBQ3V7TZJJ.json","graph_json":"https://pith.science/api/pith-number/SB7QNELSUGVFAEOSSBQ3V7TZJJ/graph.json","events_json":"https://pith.science/api/pith-number/SB7QNELSUGVFAEOSSBQ3V7TZJJ/events.json","paper":"https://pith.science/paper/SB7QNELS"},"agent_actions":{"view_html":"https://pith.science/pith/SB7QNELSUGVFAEOSSBQ3V7TZJJ","download_json":"https://pith.science/pith/SB7QNELSUGVFAEOSSBQ3V7TZJJ.json","view_paper":"https://pith.science/paper/SB7QNELS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1501.03526&json=true","fetch_graph":"https://pith.science/api/pith-number/SB7QNELSUGVFAEOSSBQ3V7TZJJ/graph.json","fetch_events":"https://pith.science/api/pith-number/SB7QNELSUGVFAEOSSBQ3V7TZJJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SB7QNELSUGVFAEOSSBQ3V7TZJJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SB7QNELSUGVFAEOSSBQ3V7TZJJ/action/storage_attestation","attest_author":"https://pith.science/pith/SB7QNELSUGVFAEOSSBQ3V7TZJJ/action/author_attestation","sign_citation":"https://pith.science/pith/SB7QNELSUGVFAEOSSBQ3V7TZJJ/action/citation_signature","submit_replication":"https://pith.science/pith/SB7QNELSUGVFAEOSSBQ3V7TZJJ/action/replication_record"}},"created_at":"2026-05-18T01:19:38.865037+00:00","updated_at":"2026-05-18T01:19:38.865037+00:00"}