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For each prime $\\ell$, there is a representation $\\rho_{E,\\ell}\\colon \\text{Gal}(\\overline{\\mathbf{Q}}/\\mathbf{Q}) \\to \\text{GL}_2(\\mathbf{F}_{\\ell})$ that describes the Galois action on the $\\ell$-torsion points of $E$. Building on recent work of Rouse--Zureick-Brown and Zywina, we find models for composite level modular curves whose rational points classify elliptic curves over $\\mathbf{Q}$ with simultaneously non-surjective, composite image of Galois. We also provably determine the rational points on almo"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1707.04646","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-07-14T21:38:03Z","cross_cats_sorted":[],"title_canon_sha256":"6902c77254fdaa727393b53ac94e878f1a792839751dce50c353669ff9d743d5","abstract_canon_sha256":"0d7382f69f408600ad8977640afd48ee341873c60cdbeecdd03a7cab0c01eb1a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:00:41.484620Z","signature_b64":"Qt3y4Gw5DUudiAX2hMxQlzKsxZwySKFTuyWTYgl/U01u4O2IGBh9mNmV9sM4Jhc9FlB29jqjA7LORHPLLI67DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7ecee8a1274b2ddd46cff97c1dee285568aa794e9a1385081a4cc0e27a1c2c39","last_reissued_at":"2026-05-18T00:00:41.484149Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:00:41.484149Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Composite images of Galois for elliptic curves over $\\mathbf{Q}$ & Entanglement fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jackson S. 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