{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:FT4R5B22Z7BDUUQSAJU5IYWHNN","short_pith_number":"pith:FT4R5B22","canonical_record":{"source":{"id":"1007.4617","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-07-27T04:27:05Z","cross_cats_sorted":[],"title_canon_sha256":"19f577fa6c406b28c23af3cb171c13419e20822a4e95b5af6025b122f5e4b6c8","abstract_canon_sha256":"7effac4192194680df233c1ba41fec44eea17a28368ae7994260282cfa938a49"},"schema_version":"1.0"},"canonical_sha256":"2cf91e875acfc23a52120269d462c76b466a8a20daa15507317f25fc77b5ae7f","source":{"kind":"arxiv","id":"1007.4617","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.4617","created_at":"2026-05-18T03:43:07Z"},{"alias_kind":"arxiv_version","alias_value":"1007.4617v3","created_at":"2026-05-18T03:43:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.4617","created_at":"2026-05-18T03:43:07Z"},{"alias_kind":"pith_short_12","alias_value":"FT4R5B22Z7BD","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"FT4R5B22Z7BDUUQS","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"FT4R5B22","created_at":"2026-05-18T12:26:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:FT4R5B22Z7BDUUQSAJU5IYWHNN","target":"record","payload":{"canonical_record":{"source":{"id":"1007.4617","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-07-27T04:27:05Z","cross_cats_sorted":[],"title_canon_sha256":"19f577fa6c406b28c23af3cb171c13419e20822a4e95b5af6025b122f5e4b6c8","abstract_canon_sha256":"7effac4192194680df233c1ba41fec44eea17a28368ae7994260282cfa938a49"},"schema_version":"1.0"},"canonical_sha256":"2cf91e875acfc23a52120269d462c76b466a8a20daa15507317f25fc77b5ae7f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:43:07.799114Z","signature_b64":"JdPWsjSUnWRU9BMbs5WHvZhdvCOo+6JmY6gF7UBohm3rlkVinjeDlV3DsH/kZwdwLW/ALt8fx8Q+3DASHjipBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2cf91e875acfc23a52120269d462c76b466a8a20daa15507317f25fc77b5ae7f","last_reissued_at":"2026-05-18T03:43:07.798299Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:43:07.798299Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1007.4617","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:43:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Iy2S53lkoDBQTwWjvK8MXycM17Per4kFhaT+VuO4sDcsVkucVNMgLO368S2k3YPmnYUphyd8kCQVYPjb2GA/Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T14:38:48.761055Z"},"content_sha256":"5e9dd90b62743aeb6c46b77ab890cf26a32b3224309276e42459245b138f9f9b","schema_version":"1.0","event_id":"sha256:5e9dd90b62743aeb6c46b77ab890cf26a32b3224309276e42459245b138f9f9b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:FT4R5B22Z7BDUUQSAJU5IYWHNN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On elliptic curves with an isogeny of degree 7","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"A. Silverberg, K. Rubin, M. Stoll, R. Greenberg","submitted_at":"2010-07-27T04:27:05Z","abstract_excerpt":"We show that if $E$ is an elliptic curve over $\\mathbf{Q}$ with a $\\mathbf{Q}$-rational isogeny of degree 7, then the image of the 7-adic Galois representation attached to $E$ is as large as allowed by the isogeny, except for the curves with complex multiplication by $\\mathbf{Q}(\\sqrt{-7})$. The analogous result with 7 replaced by a prime $p > 7$ was proved by the first author in [7]. The present case $p = 7$ has additional interesting complications. We show that any exceptions correspond to the rational points on a certain curve of genus 12. We then use the method of Chabauty to show that the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.4617","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:43:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aiNJcZ1Z2bOjpmaVusj7ZTA/frQ2huybXznFEdBgj5+kqQgmi0OP4eIz4Et88utYKqV6nH3pspA1OJxhQ1iaBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T14:38:48.761407Z"},"content_sha256":"cd64cf4936ddec2e93c5b684374bb71b0a8534e7e790fce092f9de38b461c8ee","schema_version":"1.0","event_id":"sha256:cd64cf4936ddec2e93c5b684374bb71b0a8534e7e790fce092f9de38b461c8ee"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FT4R5B22Z7BDUUQSAJU5IYWHNN/bundle.json","state_url":"https://pith.science/pith/FT4R5B22Z7BDUUQSAJU5IYWHNN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FT4R5B22Z7BDUUQSAJU5IYWHNN/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-05T14:38:48Z","links":{"resolver":"https://pith.science/pith/FT4R5B22Z7BDUUQSAJU5IYWHNN","bundle":"https://pith.science/pith/FT4R5B22Z7BDUUQSAJU5IYWHNN/bundle.json","state":"https://pith.science/pith/FT4R5B22Z7BDUUQSAJU5IYWHNN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FT4R5B22Z7BDUUQSAJU5IYWHNN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:FT4R5B22Z7BDUUQSAJU5IYWHNN","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"7effac4192194680df233c1ba41fec44eea17a28368ae7994260282cfa938a49","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-07-27T04:27:05Z","title_canon_sha256":"19f577fa6c406b28c23af3cb171c13419e20822a4e95b5af6025b122f5e4b6c8"},"schema_version":"1.0","source":{"id":"1007.4617","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.4617","created_at":"2026-05-18T03:43:07Z"},{"alias_kind":"arxiv_version","alias_value":"1007.4617v3","created_at":"2026-05-18T03:43:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.4617","created_at":"2026-05-18T03:43:07Z"},{"alias_kind":"pith_short_12","alias_value":"FT4R5B22Z7BD","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"FT4R5B22Z7BDUUQS","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"FT4R5B22","created_at":"2026-05-18T12:26:07Z"}],"graph_snapshots":[{"event_id":"sha256:cd64cf4936ddec2e93c5b684374bb71b0a8534e7e790fce092f9de38b461c8ee","target":"graph","created_at":"2026-05-18T03:43:07Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We show that if $E$ is an elliptic curve over $\\mathbf{Q}$ with a $\\mathbf{Q}$-rational isogeny of degree 7, then the image of the 7-adic Galois representation attached to $E$ is as large as allowed by the isogeny, except for the curves with complex multiplication by $\\mathbf{Q}(\\sqrt{-7})$. The analogous result with 7 replaced by a prime $p > 7$ was proved by the first author in [7]. The present case $p = 7$ has additional interesting complications. We show that any exceptions correspond to the rational points on a certain curve of genus 12. We then use the method of Chabauty to show that the","authors_text":"A. Silverberg, K. Rubin, M. Stoll, R. Greenberg","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-07-27T04:27:05Z","title":"On elliptic curves with an isogeny of degree 7"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.4617","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:5e9dd90b62743aeb6c46b77ab890cf26a32b3224309276e42459245b138f9f9b","target":"record","created_at":"2026-05-18T03:43:07Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"7effac4192194680df233c1ba41fec44eea17a28368ae7994260282cfa938a49","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-07-27T04:27:05Z","title_canon_sha256":"19f577fa6c406b28c23af3cb171c13419e20822a4e95b5af6025b122f5e4b6c8"},"schema_version":"1.0","source":{"id":"1007.4617","kind":"arxiv","version":3}},"canonical_sha256":"2cf91e875acfc23a52120269d462c76b466a8a20daa15507317f25fc77b5ae7f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2cf91e875acfc23a52120269d462c76b466a8a20daa15507317f25fc77b5ae7f","first_computed_at":"2026-05-18T03:43:07.798299Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:43:07.798299Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JdPWsjSUnWRU9BMbs5WHvZhdvCOo+6JmY6gF7UBohm3rlkVinjeDlV3DsH/kZwdwLW/ALt8fx8Q+3DASHjipBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:43:07.799114Z","signed_message":"canonical_sha256_bytes"},"source_id":"1007.4617","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5e9dd90b62743aeb6c46b77ab890cf26a32b3224309276e42459245b138f9f9b","sha256:cd64cf4936ddec2e93c5b684374bb71b0a8534e7e790fce092f9de38b461c8ee"],"state_sha256":"0b3b797afa48089cc2ea68b7e4ad03b213b37c7daa19e987771a22e08d964f3f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rSF3VCd+ypNBE3JdUa4mBsz37Gqul2hv+PcqMiMfm7gL2IVQx2ICsuLewv3FTkX3nU8XrFdkLwfKVJeg1UiIDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T14:38:48.763335Z","bundle_sha256":"2880e2ba309e0d04dfdd390dd221fdb6382bcc93f134ef98304156b2c775a1e8"}}