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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"},"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":"1404.5700","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-04-23T04:37:42Z","cross_cats_sorted":[],"title_canon_sha256":"0b3990ce7b928f9a3850b17cb6c19c2b173943870eb3fea75f9aaa6b7eb79bef","abstract_canon_sha256":"17fb78d1864c72de35b87a01bd8d18bca7f1b0ebaf5a70a68d27aa9f81b70c63"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:53:28.137530Z","signature_b64":"qAEbhfuEVpjSy+wpJH7z3lDh30P+/sKZIZ2zMtv4fUgDDjFo9N+3BXIKAMx6f/5A4EufqVu2zIWXIvxDlIjeAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f15d6742abd985fa2aaf34f246ce1f41205fe4f9bb569c829c8c35318e5af588","last_reissued_at":"2026-05-18T02:53:28.136857Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:53:28.136857Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On invariants of elliptic curves on average","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Adam Tyler Felix, Amir Akbary","submitted_at":"2014-04-23T04:37:42Z","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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.5700","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1404.5700","created_at":"2026-05-18T02:53:28.136962+00:00"},{"alias_kind":"arxiv_version","alias_value":"1404.5700v1","created_at":"2026-05-18T02:53:28.136962+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1404.5700","created_at":"2026-05-18T02:53:28.136962+00:00"},{"alias_kind":"pith_short_12","alias_value":"6FOWOQVL3GC7","created_at":"2026-05-18T12:28:16.859392+00:00"},{"alias_kind":"pith_short_16","alias_value":"6FOWOQVL3GC7UKVP","created_at":"2026-05-18T12:28:16.859392+00:00"},{"alias_kind":"pith_short_8","alias_value":"6FOWOQVL","created_at":"2026-05-18T12:28:16.859392+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/6FOWOQVL3GC7UKVPGTZENTQ7IE","json":"https://pith.science/pith/6FOWOQVL3GC7UKVPGTZENTQ7IE.json","graph_json":"https://pith.science/api/pith-number/6FOWOQVL3GC7UKVPGTZENTQ7IE/graph.json","events_json":"https://pith.science/api/pith-number/6FOWOQVL3GC7UKVPGTZENTQ7IE/events.json","paper":"https://pith.science/paper/6FOWOQVL"},"agent_actions":{"view_html":"https://pith.science/pith/6FOWOQVL3GC7UKVPGTZENTQ7IE","download_json":"https://pith.science/pith/6FOWOQVL3GC7UKVPGTZENTQ7IE.json","view_paper":"https://pith.science/paper/6FOWOQVL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1404.5700&json=true","fetch_graph":"https://pith.science/api/pith-number/6FOWOQVL3GC7UKVPGTZENTQ7IE/graph.json","fetch_events":"https://pith.science/api/pith-number/6FOWOQVL3GC7UKVPGTZENTQ7IE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6FOWOQVL3GC7UKVPGTZENTQ7IE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6FOWOQVL3GC7UKVPGTZENTQ7IE/action/storage_attestation","attest_author":"https://pith.science/pith/6FOWOQVL3GC7UKVPGTZENTQ7IE/action/author_attestation","sign_citation":"https://pith.science/pith/6FOWOQVL3GC7UKVPGTZENTQ7IE/action/citation_signature","submit_replication":"https://pith.science/pith/6FOWOQVL3GC7UKVPGTZENTQ7IE/action/replication_record"}},"created_at":"2026-05-18T02:53:28.136962+00:00","updated_at":"2026-05-18T02:53:28.136962+00:00"}