{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:4ZQ2AUXSHX4ULMIGUCCO6PZ3XP","short_pith_number":"pith:4ZQ2AUXS","canonical_record":{"source":{"id":"1107.3516","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-07-18T17:49:21Z","cross_cats_sorted":[],"title_canon_sha256":"8731d6a0b88b0b67b60e5bacc363c36d0f6b943d496abf8bff7f938d023e5c7c","abstract_canon_sha256":"814e656f819b72a3403dfd9f7bd211e100cfd4fb3ffc1f0b5b01cda1b7bfc71f"},"schema_version":"1.0"},"canonical_sha256":"e661a052f23df945b106a084ef3f3bbbdbffde4b6ca1c1dd0f4e335bee46e863","source":{"kind":"arxiv","id":"1107.3516","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.3516","created_at":"2026-05-18T01:10:11Z"},{"alias_kind":"arxiv_version","alias_value":"1107.3516v2","created_at":"2026-05-18T01:10:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.3516","created_at":"2026-05-18T01:10:11Z"},{"alias_kind":"pith_short_12","alias_value":"4ZQ2AUXSHX4U","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"4ZQ2AUXSHX4ULMIG","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"4ZQ2AUXS","created_at":"2026-05-18T12:26:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:4ZQ2AUXSHX4ULMIGUCCO6PZ3XP","target":"record","payload":{"canonical_record":{"source":{"id":"1107.3516","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-07-18T17:49:21Z","cross_cats_sorted":[],"title_canon_sha256":"8731d6a0b88b0b67b60e5bacc363c36d0f6b943d496abf8bff7f938d023e5c7c","abstract_canon_sha256":"814e656f819b72a3403dfd9f7bd211e100cfd4fb3ffc1f0b5b01cda1b7bfc71f"},"schema_version":"1.0"},"canonical_sha256":"e661a052f23df945b106a084ef3f3bbbdbffde4b6ca1c1dd0f4e335bee46e863","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:10:11.309241Z","signature_b64":"d7sR3SbN7EoFPRzrA3eR40F1TUDgM0o5GASttQyRPtabeh5gt91lUJehnKEXxwVVhJxnsgZwHAdFlVOga7pRBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e661a052f23df945b106a084ef3f3bbbdbffde4b6ca1c1dd0f4e335bee46e863","last_reissued_at":"2026-05-18T01:10:11.308770Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:10:11.308770Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1107.3516","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":"2026-05-18T01:10:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pjoAV6h3m7zV66lHHyW8kTxgKHLYKNjmbAXgqe+9yR4E4VzCnoOFQ9wcS6+hMZHb4qMEx5/ZPS/p4b5G0QaGDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T12:16:30.299589Z"},"content_sha256":"ea1cabca9220968d196262b28929d1cb75ed7384de1528a4e4ac1130bdae3e13","schema_version":"1.0","event_id":"sha256:ea1cabca9220968d196262b28929d1cb75ed7384de1528a4e4ac1130bdae3e13"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:4ZQ2AUXSHX4ULMIGUCCO6PZ3XP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Explicit n-descent on elliptic curves. III. Algorithms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Cathy O'Neil, Denis Simon, John Cremona, Michael Stoll, Tom Fisher","submitted_at":"2011-07-18T17:49:21Z","abstract_excerpt":"This is the third in a series of papers in which we study the n-Selmer group of an elliptic curve, with the aim of representing its elements as curves of degree n in P^{n-1}. The methods we describe are practical in the case n=3 for elliptic curves over the rationals, and have been implemented in Magma.\n  One important ingredient of our work is an algorithm for trivialising central simple algebras. This is of independent interest: for example, it could be used for parametrising Brauer-Severi surfaces."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.3516","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":"2026-05-18T01:10:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VCMw4Pu7yxLOV0CZRsNUxbs77caySEJakTGx3YRpq+5RI+hlM0vNfMAr2GywC6jrjprDsEyeHmDnAjB7CtO+Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T12:16:30.301949Z"},"content_sha256":"9e23045789c02059e22d36c7d179d5c077c4e97dc83643a7ac0d2014d6ce9549","schema_version":"1.0","event_id":"sha256:9e23045789c02059e22d36c7d179d5c077c4e97dc83643a7ac0d2014d6ce9549"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4ZQ2AUXSHX4ULMIGUCCO6PZ3XP/bundle.json","state_url":"https://pith.science/pith/4ZQ2AUXSHX4ULMIGUCCO6PZ3XP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4ZQ2AUXSHX4ULMIGUCCO6PZ3XP/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":"2026-05-30T12:16:30Z","links":{"resolver":"https://pith.science/pith/4ZQ2AUXSHX4ULMIGUCCO6PZ3XP","bundle":"https://pith.science/pith/4ZQ2AUXSHX4ULMIGUCCO6PZ3XP/bundle.json","state":"https://pith.science/pith/4ZQ2AUXSHX4ULMIGUCCO6PZ3XP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4ZQ2AUXSHX4ULMIGUCCO6PZ3XP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:4ZQ2AUXSHX4ULMIGUCCO6PZ3XP","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":"814e656f819b72a3403dfd9f7bd211e100cfd4fb3ffc1f0b5b01cda1b7bfc71f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-07-18T17:49:21Z","title_canon_sha256":"8731d6a0b88b0b67b60e5bacc363c36d0f6b943d496abf8bff7f938d023e5c7c"},"schema_version":"1.0","source":{"id":"1107.3516","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.3516","created_at":"2026-05-18T01:10:11Z"},{"alias_kind":"arxiv_version","alias_value":"1107.3516v2","created_at":"2026-05-18T01:10:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.3516","created_at":"2026-05-18T01:10:11Z"},{"alias_kind":"pith_short_12","alias_value":"4ZQ2AUXSHX4U","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"4ZQ2AUXSHX4ULMIG","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"4ZQ2AUXS","created_at":"2026-05-18T12:26:20Z"}],"graph_snapshots":[{"event_id":"sha256:9e23045789c02059e22d36c7d179d5c077c4e97dc83643a7ac0d2014d6ce9549","target":"graph","created_at":"2026-05-18T01:10:11Z","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":"This is the third in a series of papers in which we study the n-Selmer group of an elliptic curve, with the aim of representing its elements as curves of degree n in P^{n-1}. The methods we describe are practical in the case n=3 for elliptic curves over the rationals, and have been implemented in Magma.\n  One important ingredient of our work is an algorithm for trivialising central simple algebras. This is of independent interest: for example, it could be used for parametrising Brauer-Severi surfaces.","authors_text":"Cathy O'Neil, Denis Simon, John Cremona, Michael Stoll, Tom Fisher","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-07-18T17:49:21Z","title":"Explicit n-descent on elliptic curves. III. Algorithms"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.3516","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:ea1cabca9220968d196262b28929d1cb75ed7384de1528a4e4ac1130bdae3e13","target":"record","created_at":"2026-05-18T01:10:11Z","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":"814e656f819b72a3403dfd9f7bd211e100cfd4fb3ffc1f0b5b01cda1b7bfc71f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-07-18T17:49:21Z","title_canon_sha256":"8731d6a0b88b0b67b60e5bacc363c36d0f6b943d496abf8bff7f938d023e5c7c"},"schema_version":"1.0","source":{"id":"1107.3516","kind":"arxiv","version":2}},"canonical_sha256":"e661a052f23df945b106a084ef3f3bbbdbffde4b6ca1c1dd0f4e335bee46e863","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e661a052f23df945b106a084ef3f3bbbdbffde4b6ca1c1dd0f4e335bee46e863","first_computed_at":"2026-05-18T01:10:11.308770Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:10:11.308770Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"d7sR3SbN7EoFPRzrA3eR40F1TUDgM0o5GASttQyRPtabeh5gt91lUJehnKEXxwVVhJxnsgZwHAdFlVOga7pRBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:10:11.309241Z","signed_message":"canonical_sha256_bytes"},"source_id":"1107.3516","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ea1cabca9220968d196262b28929d1cb75ed7384de1528a4e4ac1130bdae3e13","sha256:9e23045789c02059e22d36c7d179d5c077c4e97dc83643a7ac0d2014d6ce9549"],"state_sha256":"6c23aaa57ec653a55a869d73b1538de9f3fef5f0ca846b8209cb9464b404c6a1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Z3/Z6kh3XbZqCTIVQdZYTbn4DY7nCNniHdxGUNyRnk7gI8v0vlk/3i0XXh9zhy1gR4Opgc4/coxvN0m+U9qaCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T12:16:30.306392Z","bundle_sha256":"498bca59f94e036e5dbf256f4d16cfb008cf980b44025837a0d3574895b61b99"}}