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In this note, using a recent criterion to decide if two elliptic curves over $\\mathbb{Q}$ with certain types of additive reduction at 2 have symplectically isomorphic $p$-torsion modules, we improve these densities to ${3/8}$."},"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":"1606.04374","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-06-14T13:56:29Z","cross_cats_sorted":[],"title_canon_sha256":"ca34292faef66040926afb5dd10854eaecb0fa2624c519362458cde118fa144d","abstract_canon_sha256":"d10297e8bb833701c237f9eb8d7df38b7010b70d1101c566ab0f8c12c1f48857"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:25.689535Z","signature_b64":"mvV2aFJ3faFJ7hjJdqK+fUnEmRv1Fmk55L/MAKx6VwpB4rvFXNhWLL/kTOgMMmA+XWi5aAZ4/p5sAQrlKTs+DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c9cc4cf7d9594db53704eaf43c92342c2c723bd8db05fcab5bd1044bd82d9f60","last_reissued_at":"2026-05-18T01:12:25.689147Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:25.689147Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An application of the symplectic argument to some Fermat-type Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alain Kraus, Nuno Freitas","submitted_at":"2016-06-14T13:56:29Z","abstract_excerpt":"Let $p$ be a prime number. 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