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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 $Q$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1312.1237","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-12-04T16:33:47Z","cross_cats_sorted":[],"title_canon_sha256":"6cebaea0c85029a83383feff4d580f80e62a65f524bb0e5cadd60c3b45b86c1d","abstract_canon_sha256":"2894a506f1e13a225f222a3b50e377fe767333e707840f97d8756a6d9df8114c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:18:56.249759Z","signature_b64":"HTGqpFf7PAfmvEwzxiVlWmmy+O48dJXPI6y9YJ+rBxLQXTjibhNn0VvGwl9Lq9EvH9VAKAp9cuR5CIbA2KpAAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"af8232dff71569f02c635ce743ffdfa7be9af11192e24126037cb614ad50fb87","last_reissued_at":"2026-05-18T00:18:56.249085Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:18:56.249085Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"8-rank of the class group and isotropy index","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Qing Lu","submitted_at":"2013-12-04T16:33:47Z","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. 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