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We prove that for all rank two elliptic curves with conductor smaller than 1000, the critical subgroup is torsion. First, we define a family of critical polynomials attached to $E$ and describe two algorithms to compute such polynomials. We then give a sufficient condition for the critical subgroup to be torsion in terms of the factorization of critical"},"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":"1412.2827","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-12-09T01:41:09Z","cross_cats_sorted":[],"title_canon_sha256":"3fc3b75ba8987ccf9df7917705615bf8b85ac081b166822357ef8cb4810844c5","abstract_canon_sha256":"4c9ec58b1ba8898c74374b41d249090e17603204ea31d1ca77751347ec0bd963"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:29:10.247726Z","signature_b64":"2k6WNiJDRtVlU8UQKnd6MhkQ4G/2nLbfqlIUD2ajF/hM9oJBR1bDFZ/I/a8sKagun/tjVL3+rDSweMMCa4NdCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"64a9522e7e373a35b08e4653f4a94d7a174b8f1a334ef3f7d9aead00939a1849","last_reissued_at":"2026-05-18T02:29:10.247103Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:29:10.247103Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Computing the Mazur and Swinnerton-Dyer critical subgroup of elliptic curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Hao Chen","submitted_at":"2014-12-09T01:41:09Z","abstract_excerpt":"Let $E$ be an optimal elliptic curve defined over $\\mathbb{Q}$. 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