{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:V6BDFX7XCVU7ALDDLTTUH767U6","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":"2894a506f1e13a225f222a3b50e377fe767333e707840f97d8756a6d9df8114c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-12-04T16:33:47Z","title_canon_sha256":"6cebaea0c85029a83383feff4d580f80e62a65f524bb0e5cadd60c3b45b86c1d"},"schema_version":"1.0","source":{"id":"1312.1237","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.1237","created_at":"2026-05-18T00:18:56Z"},{"alias_kind":"arxiv_version","alias_value":"1312.1237v2","created_at":"2026-05-18T00:18:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.1237","created_at":"2026-05-18T00:18:56Z"},{"alias_kind":"pith_short_12","alias_value":"V6BDFX7XCVU7","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_16","alias_value":"V6BDFX7XCVU7ALDD","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_8","alias_value":"V6BDFX7X","created_at":"2026-05-18T12:28:04Z"}],"graph_snapshots":[{"event_id":"sha256:938d2404ed717d44d8ecc7708b9eda3be634919d2c0a50ea5c31bb62b4a1ab45","target":"graph","created_at":"2026-05-18T00:18:56Z","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":"Suppose $F=\\mathbb{Q}(\\sqrt{-p_1\\dotsm p_t})$ is an imaginary quadratic number field with distinct primes $p_1,\\dots,p_{t}$, where $p_i\\equiv 1\\pmod{4}$ ($i=1,\\dots,t-1$) and $p_t\\equiv 3\\pmod{4}$. We express the possible values of the 8-rank $r_8$ of the class group of $F$ in terms of a quadratic form $Q$ over $\\mathbb{F}_2$ which is defined by quartic symbols. In particular, we show that $r_8$ is bounded by the isotropy index of $Q$.","authors_text":"Qing Lu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-12-04T16:33:47Z","title":"8-rank of the class group and isotropy index"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.1237","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:f30d0be3cf97fd7804b7c6e062032be77977426a014788f93cb7a002f63a0b01","target":"record","created_at":"2026-05-18T00:18:56Z","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":"2894a506f1e13a225f222a3b50e377fe767333e707840f97d8756a6d9df8114c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-12-04T16:33:47Z","title_canon_sha256":"6cebaea0c85029a83383feff4d580f80e62a65f524bb0e5cadd60c3b45b86c1d"},"schema_version":"1.0","source":{"id":"1312.1237","kind":"arxiv","version":2}},"canonical_sha256":"af8232dff71569f02c635ce743ffdfa7be9af11192e24126037cb614ad50fb87","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"af8232dff71569f02c635ce743ffdfa7be9af11192e24126037cb614ad50fb87","first_computed_at":"2026-05-18T00:18:56.249085Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:18:56.249085Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HTGqpFf7PAfmvEwzxiVlWmmy+O48dJXPI6y9YJ+rBxLQXTjibhNn0VvGwl9Lq9EvH9VAKAp9cuR5CIbA2KpAAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:18:56.249759Z","signed_message":"canonical_sha256_bytes"},"source_id":"1312.1237","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f30d0be3cf97fd7804b7c6e062032be77977426a014788f93cb7a002f63a0b01","sha256:938d2404ed717d44d8ecc7708b9eda3be634919d2c0a50ea5c31bb62b4a1ab45"],"state_sha256":"fbb217f9dccbd7988fcdd32f9e1409e0eedc590168f0491f55d50053b6b44400"}