{"paper":{"title":"A classification of curious Galois groups as direct products","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Garen Chiloyan","submitted_at":"2023-10-30T20:10:12Z","abstract_excerpt":"Let $N$ be a positive integer. Let $\\operatorname{H}$ be a group of level $N$ and let $E$ be an elliptic curve defined over the rationals with $\\textit{j}_{E} \\neq 0, 1728$. Then the image $\\overline{\\rho}_{E,N}\\left(\\operatorname{Gal}\\left(\\overline{\\mathbb{Q}}/\\mathbb{Q}\\right)\\right)$, of the mod-$N$ Galois representation attached to $E$, is conjugate to a subgroup of $\\operatorname{H}$ if and only if $E$ corresponds to a non-cuspidal rational point on the modular curve $\\operatorname{X}_{\\operatorname{H}}$ generated by $\\operatorname{H}$. In this article, we are interested when $\\overline{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2310.19987","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2310.19987/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"}