{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:ENJNKRQDEWOC2OBOPP7XCOW7UQ","short_pith_number":"pith:ENJNKRQD","canonical_record":{"source":{"id":"math/0606580","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2006-06-23T08:29:03Z","cross_cats_sorted":[],"title_canon_sha256":"fa4fc3c8cc7257a0fb6fe9b63c102a251dcdd2a578955ffd7f5d58cff333c843","abstract_canon_sha256":"febfd68c19fc013a50e129b27b1499ceb29d93314eb0e277428c2bea4df4a640"},"schema_version":"1.0"},"canonical_sha256":"2352d54603259c2d382e7bff713adfa404c4c14a95472d59a976814bb23718f2","source":{"kind":"arxiv","id":"math/0606580","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0606580","created_at":"2026-05-18T01:10:00Z"},{"alias_kind":"arxiv_version","alias_value":"math/0606580v1","created_at":"2026-05-18T01:10:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0606580","created_at":"2026-05-18T01:10:00Z"},{"alias_kind":"pith_short_12","alias_value":"ENJNKRQDEWOC","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"ENJNKRQDEWOC2OBO","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"ENJNKRQD","created_at":"2026-05-18T12:25:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:ENJNKRQDEWOC2OBOPP7XCOW7UQ","target":"record","payload":{"canonical_record":{"source":{"id":"math/0606580","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2006-06-23T08:29:03Z","cross_cats_sorted":[],"title_canon_sha256":"fa4fc3c8cc7257a0fb6fe9b63c102a251dcdd2a578955ffd7f5d58cff333c843","abstract_canon_sha256":"febfd68c19fc013a50e129b27b1499ceb29d93314eb0e277428c2bea4df4a640"},"schema_version":"1.0"},"canonical_sha256":"2352d54603259c2d382e7bff713adfa404c4c14a95472d59a976814bb23718f2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:10:00.371261Z","signature_b64":"jpBkhaA5OUntVzsBmt8zK0ZrxaigaEMGw3WHXumF1c3NvloN6ZGKu0U68bpyFiP1KtS9zhIoDa2PljT2Hf+0DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2352d54603259c2d382e7bff713adfa404c4c14a95472d59a976814bb23718f2","last_reissued_at":"2026-05-18T01:10:00.370517Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:10:00.370517Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0606580","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:10:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sDE5XIuQxNxf8FLy+UpUpgdCOV3a3/3upvTsdPyZZdCO1lsCO3H3NZnl2tQcNiy0MPMX5WDoJiWxD2k8yQbtAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T10:28:11.795501Z"},"content_sha256":"40fe056d2832ad431e234271e1355e964745d2587eac9dd6754af54ac773af76","schema_version":"1.0","event_id":"sha256:40fe056d2832ad431e234271e1355e964745d2587eac9dd6754af54ac773af76"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:ENJNKRQDEWOC2OBOPP7XCOW7UQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Explicit n-descent on elliptic curves, I. Algebra","license":"","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Cathy O'Neil, Denis Simon, John Cremona, Michael Stoll, Tom Fisher","submitted_at":"2006-06-23T08:29:03Z","abstract_excerpt":"This is the first 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 genus one normal curves of degree n. The methods we describe are practical in the case n=3 for elliptic curves over the rationals, and have been implemented in Magma."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0606580","kind":"arxiv","version":1},"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":"2026-05-18T01:10:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Vdclw4rjywl9kK/86SquNFF8st/veVZhpfeQfghchHd6MkTZbpvhfO7+9Oh/XPqyQtklP5XQrb3LQvmzDlGPDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T10:28:11.795867Z"},"content_sha256":"646e1e8a292e77c5984ea70161c61058cce025cc546fc5b3a24e7504e0b94465","schema_version":"1.0","event_id":"sha256:646e1e8a292e77c5984ea70161c61058cce025cc546fc5b3a24e7504e0b94465"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ENJNKRQDEWOC2OBOPP7XCOW7UQ/bundle.json","state_url":"https://pith.science/pith/ENJNKRQDEWOC2OBOPP7XCOW7UQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ENJNKRQDEWOC2OBOPP7XCOW7UQ/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":"2026-05-27T10:28:11Z","links":{"resolver":"https://pith.science/pith/ENJNKRQDEWOC2OBOPP7XCOW7UQ","bundle":"https://pith.science/pith/ENJNKRQDEWOC2OBOPP7XCOW7UQ/bundle.json","state":"https://pith.science/pith/ENJNKRQDEWOC2OBOPP7XCOW7UQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ENJNKRQDEWOC2OBOPP7XCOW7UQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:ENJNKRQDEWOC2OBOPP7XCOW7UQ","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":"febfd68c19fc013a50e129b27b1499ceb29d93314eb0e277428c2bea4df4a640","cross_cats_sorted":[],"license":"","primary_cat":"math.NT","submitted_at":"2006-06-23T08:29:03Z","title_canon_sha256":"fa4fc3c8cc7257a0fb6fe9b63c102a251dcdd2a578955ffd7f5d58cff333c843"},"schema_version":"1.0","source":{"id":"math/0606580","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0606580","created_at":"2026-05-18T01:10:00Z"},{"alias_kind":"arxiv_version","alias_value":"math/0606580v1","created_at":"2026-05-18T01:10:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0606580","created_at":"2026-05-18T01:10:00Z"},{"alias_kind":"pith_short_12","alias_value":"ENJNKRQDEWOC","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"ENJNKRQDEWOC2OBO","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"ENJNKRQD","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:646e1e8a292e77c5984ea70161c61058cce025cc546fc5b3a24e7504e0b94465","target":"graph","created_at":"2026-05-18T01:10:00Z","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 first 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 genus one normal curves of degree n. The methods we describe are practical in the case n=3 for elliptic curves over the rationals, and have been implemented in Magma.","authors_text":"Cathy O'Neil, Denis Simon, John Cremona, Michael Stoll, Tom Fisher","cross_cats":[],"headline":"","license":"","primary_cat":"math.NT","submitted_at":"2006-06-23T08:29:03Z","title":"Explicit n-descent on elliptic curves, I. Algebra"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0606580","kind":"arxiv","version":1},"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":"sha256:40fe056d2832ad431e234271e1355e964745d2587eac9dd6754af54ac773af76","target":"record","created_at":"2026-05-18T01:10:00Z","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":"febfd68c19fc013a50e129b27b1499ceb29d93314eb0e277428c2bea4df4a640","cross_cats_sorted":[],"license":"","primary_cat":"math.NT","submitted_at":"2006-06-23T08:29:03Z","title_canon_sha256":"fa4fc3c8cc7257a0fb6fe9b63c102a251dcdd2a578955ffd7f5d58cff333c843"},"schema_version":"1.0","source":{"id":"math/0606580","kind":"arxiv","version":1}},"canonical_sha256":"2352d54603259c2d382e7bff713adfa404c4c14a95472d59a976814bb23718f2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2352d54603259c2d382e7bff713adfa404c4c14a95472d59a976814bb23718f2","first_computed_at":"2026-05-18T01:10:00.370517Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:10:00.370517Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jpBkhaA5OUntVzsBmt8zK0ZrxaigaEMGw3WHXumF1c3NvloN6ZGKu0U68bpyFiP1KtS9zhIoDa2PljT2Hf+0DA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:10:00.371261Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0606580","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:40fe056d2832ad431e234271e1355e964745d2587eac9dd6754af54ac773af76","sha256:646e1e8a292e77c5984ea70161c61058cce025cc546fc5b3a24e7504e0b94465"],"state_sha256":"81a561fb748bc32af318e837abf9a708e53703bbdde7e20f0d7e43c79bd1e3d9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"w32EzivOh6wD/CaoPKkGpmWsdXeL9txJdimjnOtiDYkvPue+sMRSqMDmTeGBr8unKb9yT9BxQd3vpYfH5kW1CQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T10:28:11.798982Z","bundle_sha256":"cf2693aa937ce6853c1dc8ecf97ce8a4ea2b1dfd64a5357a4c213cf9798f3883"}}