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Let $A$ denote one of the two curves. In this paper, using Waldspurger formula and an induction method, for $n\\equiv 3,7\\mod 24$ positive square-free, as well as some other residue classes, we express the parity of analytic Sha of $A$ in terms of the genus number $g(m):=\\#2\\mathrm{Cl}(\\mathbb{Q}(\\sq"},"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":"2405.11132","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.NT","submitted_at":"2024-05-18T01:08:36Z","cross_cats_sorted":[],"title_canon_sha256":"58b9b9f85669386d605ecdedb2fe724481a5c533e38d3c1b45359f5a2f558e08","abstract_canon_sha256":"566dc46db069cbc17d0909a4eca42dae44d944ca46590374fbd758c370983ddc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:20:33.732924Z","signature_b64":"raC233g60/gnWAOXHQ1IQuVr7q5rPV1XE5ob/QaevWhGPhJjNs7UC4kmHM13nNL1tPf+QJvUNIxeYOD/WFb8DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"28af37552a791af3809eb624745a079d4320a3cc17b4098087bfcea45a8dde0f","last_reissued_at":"2026-07-05T08:20:33.732424Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:20:33.732424Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Quadratic twists of tiling number elliptic curves","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jinzhao Pan, Keqin Feng, Qiuyue Liu, Ye Tian","submitted_at":"2024-05-18T01:08:36Z","abstract_excerpt":"A positive integer $n$ is called a tiling number if the equilateral triangle can be dissected into $nk^2$ congruent triangles for some integer $k$. 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In this paper, using Waldspurger formula and an induction method, for $n\\equiv 3,7\\mod 24$ positive square-free, as well as some other residue classes, we express the parity of analytic Sha of $A$ in terms of the genus number $g(m):=\\#2\\mathrm{Cl}(\\mathbb{Q}(\\sq"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2405.11132","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2405.11132/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2405.11132","created_at":"2026-07-05T08:20:33.732484+00:00"},{"alias_kind":"arxiv_version","alias_value":"2405.11132v1","created_at":"2026-07-05T08:20:33.732484+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2405.11132","created_at":"2026-07-05T08:20:33.732484+00:00"},{"alias_kind":"pith_short_12","alias_value":"FCXTOVJKPENP","created_at":"2026-07-05T08:20:33.732484+00:00"},{"alias_kind":"pith_short_16","alias_value":"FCXTOVJKPENPHAE6","created_at":"2026-07-05T08:20:33.732484+00:00"},{"alias_kind":"pith_short_8","alias_value":"FCXTOVJK","created_at":"2026-07-05T08:20:33.732484+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/FCXTOVJKPENPHAE6WYSHIWQHTV","json":"https://pith.science/pith/FCXTOVJKPENPHAE6WYSHIWQHTV.json","graph_json":"https://pith.science/api/pith-number/FCXTOVJKPENPHAE6WYSHIWQHTV/graph.json","events_json":"https://pith.science/api/pith-number/FCXTOVJKPENPHAE6WYSHIWQHTV/events.json","paper":"https://pith.science/paper/FCXTOVJK"},"agent_actions":{"view_html":"https://pith.science/pith/FCXTOVJKPENPHAE6WYSHIWQHTV","download_json":"https://pith.science/pith/FCXTOVJKPENPHAE6WYSHIWQHTV.json","view_paper":"https://pith.science/paper/FCXTOVJK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2405.11132&json=true","fetch_graph":"https://pith.science/api/pith-number/FCXTOVJKPENPHAE6WYSHIWQHTV/graph.json","fetch_events":"https://pith.science/api/pith-number/FCXTOVJKPENPHAE6WYSHIWQHTV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FCXTOVJKPENPHAE6WYSHIWQHTV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FCXTOVJKPENPHAE6WYSHIWQHTV/action/storage_attestation","attest_author":"https://pith.science/pith/FCXTOVJKPENPHAE6WYSHIWQHTV/action/author_attestation","sign_citation":"https://pith.science/pith/FCXTOVJKPENPHAE6WYSHIWQHTV/action/citation_signature","submit_replication":"https://pith.science/pith/FCXTOVJKPENPHAE6WYSHIWQHTV/action/replication_record"}},"created_at":"2026-07-05T08:20:33.732484+00:00","updated_at":"2026-07-05T08:20:33.732484+00:00"}