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In the case when $p=2,3,5,7,13$, Ogg made predictions about the kernels of these isogenies. We show that Ogg's conjecture is not true in general. Afterwards, we propose a strategy for proving results toward Ogg's conjecture in certain situations. Finally, we discuss this strategy in detail for $N=5\\cdot 13$."},"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":"1812.05993","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-12-14T15:54:06Z","cross_cats_sorted":[],"title_canon_sha256":"a9376d18d96e2dffa4608d38feec4ed5744ae730b9584a4d28f9378ea9e6b81e","abstract_canon_sha256":"94d9d30d5e4fbdf9f5ebccf62fb4046a261acc37f41e3e20dc02537877b32e88"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:55:42.367857Z","signature_b64":"qWxe51y5mhGi5uexk8EHjWJwmjJETyBZ1FOI8Sly2Irbta85icwcgopHV+fSn0gp0/xvMzYDTKD+Nj//YtOIDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"65dad2dcd9120a2a0ebc41f379488de60d14a04d6f08013e07d16e0f36b6f610","last_reissued_at":"2026-05-17T23:55:42.367419Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:55:42.367419Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Galois extensions and a Conjecture of Ogg","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Krzysztof Klosin, Mihran Papikian","submitted_at":"2018-12-14T15:54:06Z","abstract_excerpt":"Let $N=pq$ be a product of two distinct primes. 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