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We also prove the analogous result for the sextic twists of $j$-invariant 0 curves (Mordell curves). To prove these results, we establish a general criterion for the non-triviality of the $p$-adic logarithm of Heegner points at an Eisenstein prime $p$, in terms of the relative $p$-class numbers of certain number fields and then apply this criterion"},"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":"1609.06687","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-09-21T19:08:55Z","cross_cats_sorted":[],"title_canon_sha256":"5421823ecd21c9db4c14f7a450bfa4893033cc6ebcf9655590114a71d0493b3d","abstract_canon_sha256":"2551e9b0068a81bb551f8a54a9e69f1b5ca915ddb52a99c86a8dfd717ce0fdf6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:29:31.400578Z","signature_b64":"MWXQ1ly5tUyUXwW5Br11g2UnV5BYC7T4i8rXHc3pnMEiKUtwGWhbl2TYB6gXtm8KVDOlQpq2Jyw3dTJD4FSzDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bd448a32aa0ac34fbc2b88f64472bc219a028f1df3665956d13e4b3d4fca2998","last_reissued_at":"2026-05-18T00:29:31.399909Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:29:31.399909Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Heegner points at Eisenstein primes and twists of elliptic curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Chao Li, Daniel Kriz","submitted_at":"2016-09-21T19:08:55Z","abstract_excerpt":"Given an elliptic curve $E$ over $\\mathbb{Q}$, a celebrated conjecture of Goldfeld asserts that a positive proportion of its quadratic twists should have analytic rank 0 (resp. 1). 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