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We show that for almost all $n$, the rank of $\\mathrm{Sel}_{\\phi}(E_n)$ is $0$, and the rank of $\\mathrm{Sel}_{\\hat{\\phi}}(\\hat{E}_n)$ is determined by the number of prime factors of $n$ that are congruent to $2\\bmod 3$ and the congruence class of $n\\bmod 9$."},"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":"2211.06062","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2022-11-11T08:29:08Z","cross_cats_sorted":[],"title_canon_sha256":"9371b33fae27c1135b0cf63af5a46dd87d4b12b28bc7cac85184320822781ca3","abstract_canon_sha256":"e9c45888b096687ad8e83d9aa8ed3b57aecd8a989d3fb113cb52f29add6500fd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:06:52.462711Z","signature_b64":"xlLnTsmiLen2Bci2wyIINqRCHKPksp8sS4fuWZ5A+gkhxjy+tWwDtiSKuw//SPTDF7u46oSlp70t6Qa7pHcdBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"18cedab48e407d4fb9de1a4f8e96a027dcd4cb10264c812ad5ca4868c24343bb","last_reissued_at":"2026-07-05T09:06:52.462233Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:06:52.462233Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The $3$-isogeny Selmer groups of the elliptic curves $y^2=x^3+n^2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Stephanie Chan","submitted_at":"2022-11-11T08:29:08Z","abstract_excerpt":"Consider the family of elliptic curves $E_n:y^2=x^3+n^2$, where $n$ varies over positive cubefree integers. There is a rational $3$-isogeny $\\phi$ from $E_n$ to $\\hat{E}_n:y^2=x^3-27n^2$ and a dual isogeny $\\hat{\\phi}:\\hat{E}_n\\rightarrow E_n$. We show that for almost all $n$, the rank of $\\mathrm{Sel}_{\\phi}(E_n)$ is $0$, and the rank of $\\mathrm{Sel}_{\\hat{\\phi}}(\\hat{E}_n)$ is determined by the number of prime factors of $n$ that are congruent to $2\\bmod 3$ and the congruence class of $n\\bmod 9$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2211.06062","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2211.06062/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":"2211.06062","created_at":"2026-07-05T09:06:52.462291+00:00"},{"alias_kind":"arxiv_version","alias_value":"2211.06062v2","created_at":"2026-07-05T09:06:52.462291+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2211.06062","created_at":"2026-07-05T09:06:52.462291+00:00"},{"alias_kind":"pith_short_12","alias_value":"DDHNVNEOIB6U","created_at":"2026-07-05T09:06:52.462291+00:00"},{"alias_kind":"pith_short_16","alias_value":"DDHNVNEOIB6U7OO6","created_at":"2026-07-05T09:06:52.462291+00:00"},{"alias_kind":"pith_short_8","alias_value":"DDHNVNEO","created_at":"2026-07-05T09:06:52.462291+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/DDHNVNEOIB6U7OO6DJHY5FVAE7","json":"https://pith.science/pith/DDHNVNEOIB6U7OO6DJHY5FVAE7.json","graph_json":"https://pith.science/api/pith-number/DDHNVNEOIB6U7OO6DJHY5FVAE7/graph.json","events_json":"https://pith.science/api/pith-number/DDHNVNEOIB6U7OO6DJHY5FVAE7/events.json","paper":"https://pith.science/paper/DDHNVNEO"},"agent_actions":{"view_html":"https://pith.science/pith/DDHNVNEOIB6U7OO6DJHY5FVAE7","download_json":"https://pith.science/pith/DDHNVNEOIB6U7OO6DJHY5FVAE7.json","view_paper":"https://pith.science/paper/DDHNVNEO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2211.06062&json=true","fetch_graph":"https://pith.science/api/pith-number/DDHNVNEOIB6U7OO6DJHY5FVAE7/graph.json","fetch_events":"https://pith.science/api/pith-number/DDHNVNEOIB6U7OO6DJHY5FVAE7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DDHNVNEOIB6U7OO6DJHY5FVAE7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DDHNVNEOIB6U7OO6DJHY5FVAE7/action/storage_attestation","attest_author":"https://pith.science/pith/DDHNVNEOIB6U7OO6DJHY5FVAE7/action/author_attestation","sign_citation":"https://pith.science/pith/DDHNVNEOIB6U7OO6DJHY5FVAE7/action/citation_signature","submit_replication":"https://pith.science/pith/DDHNVNEOIB6U7OO6DJHY5FVAE7/action/replication_record"}},"created_at":"2026-07-05T09:06:52.462291+00:00","updated_at":"2026-07-05T09:06:52.462291+00:00"}