{"paper":{"title":"Supersingular $j$-invariants and the Class Number of $\\mathbb{Q}(\\sqrt{-p})$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Guanju Xiao, Lixia Luo, Yingpu Deng","submitted_at":"2021-01-13T08:51:12Z","abstract_excerpt":"For a prime $p>3$, let $D$ be the discriminant of an imaginary quadratic order with $|D|< \\frac{4}{\\sqrt{3}}\\sqrt{p}$. We research the solutions of the class polynomial $H_D(X)$ mod $p$ in $\\mathbb{F}_p$ if $D$ is not a quadratic residue in $\\mathbb{F}_p$. We also discuss the common roots of different class polynomials in $\\mathbb{F}_p$. As a result, we get a deterministic algorithm (Algorithm 3) for computing the class number of $\\mathbb{Q}(\\sqrt{-p})$. The time complexity of Algorithm 3 is $O(p^{3/4+\\epsilon})$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2101.04937","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2101.04937/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"}