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We consider hypothetical degree-two \\(L\\)-functions associated to (E) that satisfy analytic continuation, a functional equation involving an arithmetic invariant \\(\\Phi(E)\\), and yield rank bounds of the form\n  \\[\n  \\operatorname{rank}(E)\\ll \\log \\Phi(E).\n  \\]\n  Using the Modularity Theorem, we show that any such invariant must satisfy\n  \\[\n  \\Phi(E)\\ge N_E.\n  \\] Thus the conductor is minimal among arithmetic invariants that"},"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":"2506.20175","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-06-25T07:03:00Z","cross_cats_sorted":[],"title_canon_sha256":"d922c96b580748f11f539fc8ae73e71597019b36b6231f616a0fb091bdf8fbf8","abstract_canon_sha256":"0f0254e021104fe4ca1c429a7430b10caeeccf0eb443e07721cdbd223becc6a2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-19T16:09:49.811438Z","signature_b64":"ipmpIQF2Wop9uZCsV2DmRNt7k4rhA/f7V+oi0S5PwAtys4luTqMEt8ZJ5uzbsAygnFpxYxXjzjHLRAerBhZfCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"96d0db7d45bcc4298607eacfb7e880f69ce6cc3e22b9dd2cca3114ff002efed0","last_reissued_at":"2026-06-19T16:09:49.811011Z","signature_status":"signed_v1","first_computed_at":"2026-06-19T16:09:49.811011Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Minimality of the Conductor for Elliptic Curve $L$-Functions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"K. 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