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We provide a complete description of the degree of the extension $\\mathbb{Q}_{\\ell}(E_p)/\\mathbb{Q}_{\\ell}$. As a consequence, we obtain a recipe to determine the discriminant ideal of the extension $\\mathbb{Q}_{\\ell}(E_p)/\\mathbb{Q}_\\ell$ in terms of standard information on $E$."},"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":"1804.07627","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-04-20T14:11:16Z","cross_cats_sorted":[],"title_canon_sha256":"d8e28a60197a1a32291d7cad15c71fd9073733663029d3e872a8f2f753b7b2a7","abstract_canon_sha256":"bba86cef44b1d56bcaa495d1e04d8c4535ac3eafc62a1a30189364574090c3ff"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:17:58.271013Z","signature_b64":"eP7qEtunHVK3+uInNewpnoVmo9lsV/Utp+xA8Bks+JBlEOYhawehXfe6aDXlbabSiJYJ+ugei/IcRtIrRDz/BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"24bac60da3d48ebd7e4e3f1f719bea3d5eeb702656d7c0c13cb8c04ef025b612","last_reissued_at":"2026-05-18T00:17:58.270404Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:17:58.270404Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the degree of the $p$-torsion field of elliptic curves over $\\mathbb{Q}_\\ell$ for $\\ell \\neq p$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alain Kraus, Nuno Freitas","submitted_at":"2018-04-20T14:11:16Z","abstract_excerpt":"Let $\\ell$ and $p \\geq 3$ be distinct prime numbers. 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