{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:6FOWOQVL3GC7UKVPGTZENTQ7IE","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":"17fb78d1864c72de35b87a01bd8d18bca7f1b0ebaf5a70a68d27aa9f81b70c63","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-04-23T04:37:42Z","title_canon_sha256":"0b3990ce7b928f9a3850b17cb6c19c2b173943870eb3fea75f9aaa6b7eb79bef"},"schema_version":"1.0","source":{"id":"1404.5700","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1404.5700","created_at":"2026-05-18T02:53:28Z"},{"alias_kind":"arxiv_version","alias_value":"1404.5700v1","created_at":"2026-05-18T02:53:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1404.5700","created_at":"2026-05-18T02:53:28Z"},{"alias_kind":"pith_short_12","alias_value":"6FOWOQVL3GC7","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_16","alias_value":"6FOWOQVL3GC7UKVP","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_8","alias_value":"6FOWOQVL","created_at":"2026-05-18T12:28:16Z"}],"graph_snapshots":[{"event_id":"sha256:e7965e79f630753ee593773c765d605d8d98559bc2232a4660a42a119c1d0030","target":"graph","created_at":"2026-05-18T02:53:28Z","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 prove several results regarding some invariants of elliptic curves on average over the family of all elliptic curves inside a box of sides $A$ and $B$. As an example, let $E$ be an elliptic curve defined over $\\mathbb{Q}$ and $p$ be a prime of good reduction for $E$. Let $e_{E}(p)$ be the exponent of the group of rational points of the reduction modulo $p$ of $E$ over the finite field $\\mathbb{F}_p$. Let $\\mathcal{C}$ be the family of elliptic curves $$E_{a,b}:~y^2=x^3+ax+b,$$ where $|a|\\leq A$ and $|b|\\leq B$. We prove that, for any $c>1$ and $k\\in \\mathbb{N}$, $$\\frac{1}{|\\mathcal{C}|} \\s","authors_text":"Adam Tyler Felix, Amir Akbary","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-04-23T04:37:42Z","title":"On invariants of elliptic curves on average"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.5700","kind":"arxiv","version":1},"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":"sha256:3860474d70a4ed31f592061837a11ccba4b1357057f1fa272c42cf98348b9154","target":"record","created_at":"2026-05-18T02:53:28Z","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":"17fb78d1864c72de35b87a01bd8d18bca7f1b0ebaf5a70a68d27aa9f81b70c63","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-04-23T04:37:42Z","title_canon_sha256":"0b3990ce7b928f9a3850b17cb6c19c2b173943870eb3fea75f9aaa6b7eb79bef"},"schema_version":"1.0","source":{"id":"1404.5700","kind":"arxiv","version":1}},"canonical_sha256":"f15d6742abd985fa2aaf34f246ce1f41205fe4f9bb569c829c8c35318e5af588","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f15d6742abd985fa2aaf34f246ce1f41205fe4f9bb569c829c8c35318e5af588","first_computed_at":"2026-05-18T02:53:28.136857Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:53:28.136857Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qAEbhfuEVpjSy+wpJH7z3lDh30P+/sKZIZ2zMtv4fUgDDjFo9N+3BXIKAMx6f/5A4EufqVu2zIWXIvxDlIjeAg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:53:28.137530Z","signed_message":"canonical_sha256_bytes"},"source_id":"1404.5700","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3860474d70a4ed31f592061837a11ccba4b1357057f1fa272c42cf98348b9154","sha256:e7965e79f630753ee593773c765d605d8d98559bc2232a4660a42a119c1d0030"],"state_sha256":"77add74da9f1c2b6d60da42efaf48fb469408e4b3982b1fed0d42b2f277f33f3"}