{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:FXFSZ5CXPNFPCV4JHBAAF4EPLV","short_pith_number":"pith:FXFSZ5CX","canonical_record":{"source":{"id":"2002.04134","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-02-10T23:36:30Z","cross_cats_sorted":[],"title_canon_sha256":"d762b28b0c8ddf634e56c2dc27b4feaa23019cbf8c0efa307ef6bee88916f776","abstract_canon_sha256":"a5049bda802f10aca934e6308d013e1d06001c1ced7c29f9805c48b653d223a5"},"schema_version":"1.0"},"canonical_sha256":"2dcb2cf4577b4af15789384002f08f5d56db0af654a39a8e40795f52dd791a44","source":{"kind":"arxiv","id":"2002.04134","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2002.04134","created_at":"2026-07-05T02:23:04Z"},{"alias_kind":"arxiv_version","alias_value":"2002.04134v4","created_at":"2026-07-05T02:23:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2002.04134","created_at":"2026-07-05T02:23:04Z"},{"alias_kind":"pith_short_12","alias_value":"FXFSZ5CXPNFP","created_at":"2026-07-05T02:23:04Z"},{"alias_kind":"pith_short_16","alias_value":"FXFSZ5CXPNFPCV4J","created_at":"2026-07-05T02:23:04Z"},{"alias_kind":"pith_short_8","alias_value":"FXFSZ5CX","created_at":"2026-07-05T02:23:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:FXFSZ5CXPNFPCV4JHBAAF4EPLV","target":"record","payload":{"canonical_record":{"source":{"id":"2002.04134","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-02-10T23:36:30Z","cross_cats_sorted":[],"title_canon_sha256":"d762b28b0c8ddf634e56c2dc27b4feaa23019cbf8c0efa307ef6bee88916f776","abstract_canon_sha256":"a5049bda802f10aca934e6308d013e1d06001c1ced7c29f9805c48b653d223a5"},"schema_version":"1.0"},"canonical_sha256":"2dcb2cf4577b4af15789384002f08f5d56db0af654a39a8e40795f52dd791a44","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:23:04.829784Z","signature_b64":"IxnxjuQzTDlSsrvXA65Ou5KyRSmnJwf56z4d17snkL+NFT+/XFKviCXLVO7rxomH596ZKLZQtd8IE7gH5vOZAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2dcb2cf4577b4af15789384002f08f5d56db0af654a39a8e40795f52dd791a44","last_reissued_at":"2026-07-05T02:23:04.829422Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:23:04.829422Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2002.04134","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T02:23:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rGF7zLJbJkZPtL0p0SSnsN0+a8Uiq++mLMhAWqi24ElZZcXntVrq7wyF+enhOBxf0eTWgqhFYKPoAvMYsOqpCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-05T15:54:59.569895Z"},"content_sha256":"6de5a2436ccfe158aac113527ad586b50043f3bdb06959b367d57b2a70a4af1e","schema_version":"1.0","event_id":"sha256:6de5a2436ccfe158aac113527ad586b50043f3bdb06959b367d57b2a70a4af1e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:FXFSZ5CXPNFPCV4JHBAAF4EPLV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Hasse invariant of the Tate normal form $E_5$ and the class number of $\\mathbb{Q}(\\sqrt{-5l})$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Patrick Morton","submitted_at":"2020-02-10T23:36:30Z","abstract_excerpt":"It is shown that the number of irreducible quartic factors of the form $g(x) = x^4+ax^3+(11a+2)x^2-ax+1$ which divide the Hasse invariant of the Tate normal form $E_5$ in characteristic $l$ is a simple linear function of the class number $h(-5l)$ of the field $\\mathbb{Q}(\\sqrt{-5l})$, when $l \\equiv 2,3$ modulo $5$. A similar result holds for irreducible quadratic factors of $g(x)$, when $l \\equiv 1, 4$ modulo $5$. This implies a formula for the number of linear factors over $\\mathbb{F}_p$ of the supersingular polynomial $ss_p^{(5*)}(x)$ corresponding to the Fricke group $\\Gamma_0^*(5)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2002.04134","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2002.04134/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T02:23:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LmrKWIH23m8vUbqcxy3K8gs3a5E5jw3HoH73QVp9koauXYMsAsCAN1Z5o7Dy6dNdEN36t0jeE+twYqxGGz6BBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-05T15:54:59.570290Z"},"content_sha256":"6530479f52ee9d360ea583d49e7680b05875dbbcd787f1fada0cee1e59803011","schema_version":"1.0","event_id":"sha256:6530479f52ee9d360ea583d49e7680b05875dbbcd787f1fada0cee1e59803011"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FXFSZ5CXPNFPCV4JHBAAF4EPLV/bundle.json","state_url":"https://pith.science/pith/FXFSZ5CXPNFPCV4JHBAAF4EPLV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FXFSZ5CXPNFPCV4JHBAAF4EPLV/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":"2026-07-05T15:54:59Z","links":{"resolver":"https://pith.science/pith/FXFSZ5CXPNFPCV4JHBAAF4EPLV","bundle":"https://pith.science/pith/FXFSZ5CXPNFPCV4JHBAAF4EPLV/bundle.json","state":"https://pith.science/pith/FXFSZ5CXPNFPCV4JHBAAF4EPLV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FXFSZ5CXPNFPCV4JHBAAF4EPLV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:FXFSZ5CXPNFPCV4JHBAAF4EPLV","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":"a5049bda802f10aca934e6308d013e1d06001c1ced7c29f9805c48b653d223a5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-02-10T23:36:30Z","title_canon_sha256":"d762b28b0c8ddf634e56c2dc27b4feaa23019cbf8c0efa307ef6bee88916f776"},"schema_version":"1.0","source":{"id":"2002.04134","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2002.04134","created_at":"2026-07-05T02:23:04Z"},{"alias_kind":"arxiv_version","alias_value":"2002.04134v4","created_at":"2026-07-05T02:23:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2002.04134","created_at":"2026-07-05T02:23:04Z"},{"alias_kind":"pith_short_12","alias_value":"FXFSZ5CXPNFP","created_at":"2026-07-05T02:23:04Z"},{"alias_kind":"pith_short_16","alias_value":"FXFSZ5CXPNFPCV4J","created_at":"2026-07-05T02:23:04Z"},{"alias_kind":"pith_short_8","alias_value":"FXFSZ5CX","created_at":"2026-07-05T02:23:04Z"}],"graph_snapshots":[{"event_id":"sha256:6530479f52ee9d360ea583d49e7680b05875dbbcd787f1fada0cee1e59803011","target":"graph","created_at":"2026-07-05T02:23:04Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2002.04134/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"It is shown that the number of irreducible quartic factors of the form $g(x) = x^4+ax^3+(11a+2)x^2-ax+1$ which divide the Hasse invariant of the Tate normal form $E_5$ in characteristic $l$ is a simple linear function of the class number $h(-5l)$ of the field $\\mathbb{Q}(\\sqrt{-5l})$, when $l \\equiv 2,3$ modulo $5$. A similar result holds for irreducible quadratic factors of $g(x)$, when $l \\equiv 1, 4$ modulo $5$. This implies a formula for the number of linear factors over $\\mathbb{F}_p$ of the supersingular polynomial $ss_p^{(5*)}(x)$ corresponding to the Fricke group $\\Gamma_0^*(5)$.","authors_text":"Patrick Morton","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-02-10T23:36:30Z","title":"The Hasse invariant of the Tate normal form $E_5$ and the class number of $\\mathbb{Q}(\\sqrt{-5l})$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2002.04134","kind":"arxiv","version":4},"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":"sha256:6de5a2436ccfe158aac113527ad586b50043f3bdb06959b367d57b2a70a4af1e","target":"record","created_at":"2026-07-05T02:23:04Z","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":"a5049bda802f10aca934e6308d013e1d06001c1ced7c29f9805c48b653d223a5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-02-10T23:36:30Z","title_canon_sha256":"d762b28b0c8ddf634e56c2dc27b4feaa23019cbf8c0efa307ef6bee88916f776"},"schema_version":"1.0","source":{"id":"2002.04134","kind":"arxiv","version":4}},"canonical_sha256":"2dcb2cf4577b4af15789384002f08f5d56db0af654a39a8e40795f52dd791a44","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2dcb2cf4577b4af15789384002f08f5d56db0af654a39a8e40795f52dd791a44","first_computed_at":"2026-07-05T02:23:04.829422Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:23:04.829422Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IxnxjuQzTDlSsrvXA65Ou5KyRSmnJwf56z4d17snkL+NFT+/XFKviCXLVO7rxomH596ZKLZQtd8IE7gH5vOZAw==","signature_status":"signed_v1","signed_at":"2026-07-05T02:23:04.829784Z","signed_message":"canonical_sha256_bytes"},"source_id":"2002.04134","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6de5a2436ccfe158aac113527ad586b50043f3bdb06959b367d57b2a70a4af1e","sha256:6530479f52ee9d360ea583d49e7680b05875dbbcd787f1fada0cee1e59803011"],"state_sha256":"ad06f3359e2a44cfcdd740cc5c92dae8eff338be3f02a754b4ad0c179574100a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IBWQgsmg3+0TodVPDkxjN0MJAoSu1r/UJiKuWgJ0q+T0cmM6h4F2M4nvtUa94P+nC2zFwOfBZvu10fEXDPeXDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-05T15:54:59.572579Z","bundle_sha256":"15e861d6745ff32bc6979ffecc8df21133fed7179ff61e3fc2304a97e636a609"}}