{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:MSUVELT6G45DLMEOIZJ7JKKNPI","short_pith_number":"pith:MSUVELT6","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"},"canonical_sha256":"64a9522e7e373a35b08e4653f4a94d7a174b8f1a334ef3f7d9aead00939a1849","source":{"kind":"arxiv","id":"1412.2827","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.2827","created_at":"2026-05-18T02:29:10Z"},{"alias_kind":"arxiv_version","alias_value":"1412.2827v2","created_at":"2026-05-18T02:29:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.2827","created_at":"2026-05-18T02:29:10Z"},{"alias_kind":"pith_short_12","alias_value":"MSUVELT6G45D","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_16","alias_value":"MSUVELT6G45DLMEO","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_8","alias_value":"MSUVELT6","created_at":"2026-05-18T12:28:38Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:MSUVELT6G45DLMEOIZJ7JKKNPI","target":"record","payload":{"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"},"canonical_sha256":"64a9522e7e373a35b08e4653f4a94d7a174b8f1a334ef3f7d9aead00939a1849","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"},"source_kind":"arxiv","source_id":"1412.2827","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:29:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3iVvnCtUeVzkRgjDoUq8mWjSv0sFS4DIcDmeb763nhTjcyk5pQQ9tPZyCxj/P0q4SgahOcmJA8w+dvt/ucSjBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T01:35:15.752448Z"},"content_sha256":"cef8e8d54796e75564f1d061d39a0b67164c43e234b88b172d57b5f06cf82f89","schema_version":"1.0","event_id":"sha256:cef8e8d54796e75564f1d061d39a0b67164c43e234b88b172d57b5f06cf82f89"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:MSUVELT6G45DLMEOIZJ7JKKNPI","target":"graph","payload":{"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 $\\mathbb{Q}$. The critical subgroup of $E$ is defined by Mazur and Swinnerton-Dyer as the subgroup of $E(\\mathbb{Q})$ generated by traces of branch points under a modular parametrization of $E$. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.2827","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:29:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HrjsUAC+Dxcmc/T+Vm8+Y0Tsdz1bak45rF+9OipQ5DDuaxKG03eIFD7iFxUpZX2WCkbAq4RDyy2nDkunmK2tBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T01:35:15.752798Z"},"content_sha256":"f6c8af90d847d6b67b826b148d4e60c07a8266a3a6f3da001ae35099c8a21cc7","schema_version":"1.0","event_id":"sha256:f6c8af90d847d6b67b826b148d4e60c07a8266a3a6f3da001ae35099c8a21cc7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MSUVELT6G45DLMEOIZJ7JKKNPI/bundle.json","state_url":"https://pith.science/pith/MSUVELT6G45DLMEOIZJ7JKKNPI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MSUVELT6G45DLMEOIZJ7JKKNPI/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-07T01:35:15Z","links":{"resolver":"https://pith.science/pith/MSUVELT6G45DLMEOIZJ7JKKNPI","bundle":"https://pith.science/pith/MSUVELT6G45DLMEOIZJ7JKKNPI/bundle.json","state":"https://pith.science/pith/MSUVELT6G45DLMEOIZJ7JKKNPI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MSUVELT6G45DLMEOIZJ7JKKNPI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:MSUVELT6G45DLMEOIZJ7JKKNPI","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"4c9ec58b1ba8898c74374b41d249090e17603204ea31d1ca77751347ec0bd963","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-12-09T01:41:09Z","title_canon_sha256":"3fc3b75ba8987ccf9df7917705615bf8b85ac081b166822357ef8cb4810844c5"},"schema_version":"1.0","source":{"id":"1412.2827","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.2827","created_at":"2026-05-18T02:29:10Z"},{"alias_kind":"arxiv_version","alias_value":"1412.2827v2","created_at":"2026-05-18T02:29:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.2827","created_at":"2026-05-18T02:29:10Z"},{"alias_kind":"pith_short_12","alias_value":"MSUVELT6G45D","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_16","alias_value":"MSUVELT6G45DLMEO","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_8","alias_value":"MSUVELT6","created_at":"2026-05-18T12:28:38Z"}],"graph_snapshots":[{"event_id":"sha256:f6c8af90d847d6b67b826b148d4e60c07a8266a3a6f3da001ae35099c8a21cc7","target":"graph","created_at":"2026-05-18T02:29:10Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $E$ be an optimal elliptic curve defined over $\\mathbb{Q}$. The critical subgroup of $E$ is defined by Mazur and Swinnerton-Dyer as the subgroup of $E(\\mathbb{Q})$ generated by traces of branch points under a modular parametrization of $E$. 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","authors_text":"Hao Chen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-12-09T01:41:09Z","title":"Computing the Mazur and Swinnerton-Dyer critical subgroup of elliptic curves"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.2827","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:cef8e8d54796e75564f1d061d39a0b67164c43e234b88b172d57b5f06cf82f89","target":"record","created_at":"2026-05-18T02:29:10Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"4c9ec58b1ba8898c74374b41d249090e17603204ea31d1ca77751347ec0bd963","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-12-09T01:41:09Z","title_canon_sha256":"3fc3b75ba8987ccf9df7917705615bf8b85ac081b166822357ef8cb4810844c5"},"schema_version":"1.0","source":{"id":"1412.2827","kind":"arxiv","version":2}},"canonical_sha256":"64a9522e7e373a35b08e4653f4a94d7a174b8f1a334ef3f7d9aead00939a1849","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"64a9522e7e373a35b08e4653f4a94d7a174b8f1a334ef3f7d9aead00939a1849","first_computed_at":"2026-05-18T02:29:10.247103Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:29:10.247103Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2k6WNiJDRtVlU8UQKnd6MhkQ4G/2nLbfqlIUD2ajF/hM9oJBR1bDFZ/I/a8sKagun/tjVL3+rDSweMMCa4NdCw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:29:10.247726Z","signed_message":"canonical_sha256_bytes"},"source_id":"1412.2827","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cef8e8d54796e75564f1d061d39a0b67164c43e234b88b172d57b5f06cf82f89","sha256:f6c8af90d847d6b67b826b148d4e60c07a8266a3a6f3da001ae35099c8a21cc7"],"state_sha256":"52d0eb6c1ef2ec1f765b62a22a4375e041e96aee4c763f6ff332c2002e37da26"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0YmJ2zQMGUZ8cMdKHpgVZINt42vdPVnqdFxxsPUtdyj5HrkzQIl3KC4zdiBacx8jRhuiWhckwajIh4ikIwhUDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T01:35:15.756154Z","bundle_sha256":"1ce9474bbeb03421eb6349aa500ec7bc794ae84e928da6c781e36d73fd38a64c"}}