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We give a lower bounds in terms of the Mordell-Weil rank of $E(\\Q)$. As an application of our result, we give an example such that p^{2n} divides the class number of the field $K_n$ in the case of $p=5077$ for each positive integer n."},"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":"1307.7691","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-07-29T19:24:31Z","cross_cats_sorted":[],"title_canon_sha256":"956545abb98bb0057d19a91ac497fc5b8c94970f3726ef52127606242797859f","abstract_canon_sha256":"03330697bfd799194ebcb1c08c9153acb0e95ad05fca393116c9f41cb98b9941"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:55:58.972488Z","signature_b64":"6HM70DuUzoiXf04YeNTZQLfzGaKXwR0c2ugomLVlsVnFzYqSP2FMSxJ/I+w0eAhchbfFEqoBHdcX3G4GHLykAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"82470febecad75c8bf784dcaffad2e80636c1dc1b328efccb7437afa60ce020e","last_reissued_at":"2026-05-18T02:55:58.971928Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:55:58.971928Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the class numbers of the fields of the p^n-torsion points of certain elliptic curves over Q","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Fumio Sairaiji, Takuya Yamauchi","submitted_at":"2013-07-29T19:24:31Z","abstract_excerpt":"Let E be an elliptic curve over Q with prime conductor p. 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