{"paper":{"title":"Spins of prime ideals and the negative Pell equation $x^2 - 2py^2 = -1$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Djordjo Milovic, Peter Koymans","submitted_at":"2016-11-30T20:06:45Z","abstract_excerpt":"Let $p\\equiv 1\\bmod 4$ be a prime number. We use a number field variant of Vinogradov's method to prove density results about the following four arithmetic invariants: (i) $16$-rank of the class group $\\mathrm{Cl}(-4p)$ of the imaginary quadratic number field $\\mathbb{Q}(\\sqrt{-4p})$; (ii) $8$-rank of the ordinary class group $\\mathrm{Cl}(8p)$ of the real quadratic field $\\mathbb{Q}(\\sqrt{8p})$; (iii) the solvability of the negative Pell equation $x^2 - 2py^2 = -1$ over the integers; (iv) $2$-part of the Tate-\\v{S}afarevi\\v{c} group of the congruent number elliptic curve $E_p: y^2 = x^3-p^2x$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.10337","kind":"arxiv","version":3},"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"}