{"paper":{"title":"On the Selmer group and rank of a family of elliptic curves and curves of genus one violating the Hasse principle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Eleni Agathocleous","submitted_at":"2021-11-05T20:40:46Z","abstract_excerpt":"We study an infinite family of $j$-invariant zero elliptic curves $E_{D}:y^{2}=x^{3}+16D$ and their $\\lambda$-isogenous curves $E_{D'}:y^{2}=x^{3}-27\\cdot16D$, where $D$ and $D' = -3D$ are fundamental discriminants of a specific form, and $\\lambda$ is an isogeny of degree $3$. A result of Honda guarantees that for our discriminants $D$, the quadratic number field $K_{D} = \\mathbb{Q}(\\sqrt{D})$ always has non-trivial 3-class group. We prove a series of results related to the set of rational points $E_{D'}(\\mathbb{Q}) \\setminus \\lambda(E_{D}(\\mathbb{Q}))$, and the $SL(2,\\mathbb{Z})$-equivalence "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2111.03723","kind":"arxiv","version":7},"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/2111.03723/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"}