{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:52UJT6VQXD52HTJ5Q743KDAU53","short_pith_number":"pith:52UJT6VQ","canonical_record":{"source":{"id":"1001.0402","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-01-03T19:46:54Z","cross_cats_sorted":[],"title_canon_sha256":"774c105af36ff66bfe5328f7d423ed13534e74dba69dd515e4a39534575407f2","abstract_canon_sha256":"5826033c9b788f562bfd49e22fb26131f28410c5f15f927fc82f6c6c1c74b44b"},"schema_version":"1.0"},"canonical_sha256":"eea899fab0b8fba3cd3d87f9b50c14eec67e74d091fb23e768b45a164ae2e4bd","source":{"kind":"arxiv","id":"1001.0402","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1001.0402","created_at":"2026-05-18T03:34:46Z"},{"alias_kind":"arxiv_version","alias_value":"1001.0402v2","created_at":"2026-05-18T03:34:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1001.0402","created_at":"2026-05-18T03:34:46Z"},{"alias_kind":"pith_short_12","alias_value":"52UJT6VQXD52","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"52UJT6VQXD52HTJ5","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"52UJT6VQ","created_at":"2026-05-18T12:26:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:52UJT6VQXD52HTJ5Q743KDAU53","target":"record","payload":{"canonical_record":{"source":{"id":"1001.0402","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-01-03T19:46:54Z","cross_cats_sorted":[],"title_canon_sha256":"774c105af36ff66bfe5328f7d423ed13534e74dba69dd515e4a39534575407f2","abstract_canon_sha256":"5826033c9b788f562bfd49e22fb26131f28410c5f15f927fc82f6c6c1c74b44b"},"schema_version":"1.0"},"canonical_sha256":"eea899fab0b8fba3cd3d87f9b50c14eec67e74d091fb23e768b45a164ae2e4bd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:34:46.658195Z","signature_b64":"A9wDA4pS9UxduJuoVnzd0Ysidgo9NjGyvCzo5FJwBlpciirbgBaSu8GA23BBsxFnm9xQ8ytU6x5SSlhEwuudDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eea899fab0b8fba3cd3d87f9b50c14eec67e74d091fb23e768b45a164ae2e4bd","last_reissued_at":"2026-05-18T03:34:46.657544Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:34:46.657544Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1001.0402","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":"2026-05-18T03:34:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"shcmxvpSKcUBuGUf2lvMVe/aUs02E4v/n+i0Y9+/RJDh2O17vW7aE6WfxBBLv0Mc6e/oxrTKjaaooCs0n2Y6BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T20:51:57.772302Z"},"content_sha256":"42526200dca34fdaed232cb5243cc8ab56103d86e6be4336133b5c4e8c96ebac","schema_version":"1.0","event_id":"sha256:42526200dca34fdaed232cb5243cc8ab56103d86e6be4336133b5c4e8c96ebac"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:52UJT6VQXD52HTJ5Q743KDAU53","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Modular polynomials via isogeny volcanoes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andrew V. Sutherland, Kristin Lauter, Reinier Broker","submitted_at":"2010-01-03T19:46:54Z","abstract_excerpt":"We present a new algorithm to compute the classical modular polynomial Phi_n in the rings Z[X,Y] and (Z/mZ)[X,Y], for a prime n and any positive integer m. Our approach uses the graph of n-isogenies to efficiently compute Phi_n mod p for many primes p of a suitable form, and then applies the Chinese Remainder Theorem (CRT). Under the Generalized Riemann Hypothesis (GRH), we achieve an expected running time of O(n^3 (log n)^3 log log n), and compute Phi_n mod m using O(n^2 (log n)^2 + n^2 log m) space. We have used the new algorithm to compute Phi_n with n over 5000, and Phi_n mod m with n over"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.0402","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":"2026-05-18T03:34:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ueVfTE6EiZsOW5MImm0pMQcVvBcKlxIYTVjqWiVtXiDvHT/gwyiGs9nCD/OYS9oVwua4odGPekIoFsLEJkEvAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T20:51:57.772681Z"},"content_sha256":"11e0a3f3ced6c2571d9ebf850f29a5e551916247da8537962ac7c172915b0b8b","schema_version":"1.0","event_id":"sha256:11e0a3f3ced6c2571d9ebf850f29a5e551916247da8537962ac7c172915b0b8b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/52UJT6VQXD52HTJ5Q743KDAU53/bundle.json","state_url":"https://pith.science/pith/52UJT6VQXD52HTJ5Q743KDAU53/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/52UJT6VQXD52HTJ5Q743KDAU53/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":"2026-06-02T20:51:57Z","links":{"resolver":"https://pith.science/pith/52UJT6VQXD52HTJ5Q743KDAU53","bundle":"https://pith.science/pith/52UJT6VQXD52HTJ5Q743KDAU53/bundle.json","state":"https://pith.science/pith/52UJT6VQXD52HTJ5Q743KDAU53/state.json","well_known_bundle":"https://pith.science/.well-known/pith/52UJT6VQXD52HTJ5Q743KDAU53/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:52UJT6VQXD52HTJ5Q743KDAU53","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":"5826033c9b788f562bfd49e22fb26131f28410c5f15f927fc82f6c6c1c74b44b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-01-03T19:46:54Z","title_canon_sha256":"774c105af36ff66bfe5328f7d423ed13534e74dba69dd515e4a39534575407f2"},"schema_version":"1.0","source":{"id":"1001.0402","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1001.0402","created_at":"2026-05-18T03:34:46Z"},{"alias_kind":"arxiv_version","alias_value":"1001.0402v2","created_at":"2026-05-18T03:34:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1001.0402","created_at":"2026-05-18T03:34:46Z"},{"alias_kind":"pith_short_12","alias_value":"52UJT6VQXD52","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"52UJT6VQXD52HTJ5","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"52UJT6VQ","created_at":"2026-05-18T12:26:04Z"}],"graph_snapshots":[{"event_id":"sha256:11e0a3f3ced6c2571d9ebf850f29a5e551916247da8537962ac7c172915b0b8b","target":"graph","created_at":"2026-05-18T03:34:46Z","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":"We present a new algorithm to compute the classical modular polynomial Phi_n in the rings Z[X,Y] and (Z/mZ)[X,Y], for a prime n and any positive integer m. Our approach uses the graph of n-isogenies to efficiently compute Phi_n mod p for many primes p of a suitable form, and then applies the Chinese Remainder Theorem (CRT). Under the Generalized Riemann Hypothesis (GRH), we achieve an expected running time of O(n^3 (log n)^3 log log n), and compute Phi_n mod m using O(n^2 (log n)^2 + n^2 log m) space. We have used the new algorithm to compute Phi_n with n over 5000, and Phi_n mod m with n over","authors_text":"Andrew V. Sutherland, Kristin Lauter, Reinier Broker","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-01-03T19:46:54Z","title":"Modular polynomials via isogeny volcanoes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.0402","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":"sha256:42526200dca34fdaed232cb5243cc8ab56103d86e6be4336133b5c4e8c96ebac","target":"record","created_at":"2026-05-18T03:34:46Z","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":"5826033c9b788f562bfd49e22fb26131f28410c5f15f927fc82f6c6c1c74b44b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-01-03T19:46:54Z","title_canon_sha256":"774c105af36ff66bfe5328f7d423ed13534e74dba69dd515e4a39534575407f2"},"schema_version":"1.0","source":{"id":"1001.0402","kind":"arxiv","version":2}},"canonical_sha256":"eea899fab0b8fba3cd3d87f9b50c14eec67e74d091fb23e768b45a164ae2e4bd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eea899fab0b8fba3cd3d87f9b50c14eec67e74d091fb23e768b45a164ae2e4bd","first_computed_at":"2026-05-18T03:34:46.657544Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:34:46.657544Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"A9wDA4pS9UxduJuoVnzd0Ysidgo9NjGyvCzo5FJwBlpciirbgBaSu8GA23BBsxFnm9xQ8ytU6x5SSlhEwuudDA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:34:46.658195Z","signed_message":"canonical_sha256_bytes"},"source_id":"1001.0402","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:42526200dca34fdaed232cb5243cc8ab56103d86e6be4336133b5c4e8c96ebac","sha256:11e0a3f3ced6c2571d9ebf850f29a5e551916247da8537962ac7c172915b0b8b"],"state_sha256":"5f3fc23ab330515bdc1b820f0ce93dc1208d2b7135ec942c433985a5c737686e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VfNyi2P1+ePlKm7bU4gsKLb5OZ0x/clWrmRwrG3earqQUPn6b9RXr/UwgQuEvarWkA47A6OhziTTHytb8c6XDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T20:51:57.775085Z","bundle_sha256":"d24fe8c96bbac0a07d4cb2cbdb09a69210d420080dbfa39db318a8bda870c500"}}