{"paper":{"title":"The Group Structure of Bachet Elliptic Curves over Finite Fields F_{p}","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Gokhan Soydan, Ismail Naci Cangul, Musa Demirci, Nazli Yildiz Ikikardes","submitted_at":"2011-06-29T07:07:58Z","abstract_excerpt":"Bachet elliptic curves are the curves y^2=x^3+a^3 and in this work the group structure E(F_{p}) of these curves over finite fields F_{p} is considered. It is shown that there are two possible structures E(F_{p}){\\cong}C_{p+1} or E(F_{p}){\\cong}C_{n}{\\times}C_{nm}, for m,n{\\in}{\\mathbb{N}}, according to p{\\equiv}5 (mod6) and p{\\equiv}1 (mod6), respectively. A result of Washington is restated in a more specific way saying that if E(F_{p}){\\cong}Z_{n}{\\times}Z_{n}, then p{\\equiv}7 (mod12) and p=n^2{\\mp}n+1."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.5851","kind":"arxiv","version":1},"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"}