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As usual define the trace of Frobenius $a_{p,\\,a,\\,b}$ by \\begin{equation*}\n  \\#E_{a,b}(\\mathbb{F}_{p}) = p+1 -a_{p,\\,a,\\,b}. \\end{equation*} We use elementary facts about exponential sums and known results about binary quadratic forms over finite fields to evaluate the sums $\\sum_{t\\in\\mathbb{F}_{p}} a_{p,\\, t,\\, b}$, $\\sum _{t \\in \\mathbb{F}_{p}} a_{p,\\,a,\\, t}$,\n  $ \\sum_{t=0}^{p-1}a_{p,\\,t,\\,b}^{2}$, $ \\sum_{t=0}^{p-1}a_{p,\\,a,\\,t}^{2}"},"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":"1901.00604","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-01-03T03:59:30Z","cross_cats_sorted":[],"title_canon_sha256":"e5cac76622672c375e3caf98318dea0b1e8f50f5d56a1261f3463f9352876005","abstract_canon_sha256":"2d7544a517bdf2a291670ca8e0276e88ea7f29122f6f97e16038b54776082204"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:57:01.951159Z","signature_b64":"1/RuMR+ElIh2evOQFHBQv5I7hvlXS/qKnMTqlUxHfNwP+JXXPBgnUgkfcvkkNBsD6GfprdkkH9DQg121U1MBCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4e2404a8f4f5caf35ec33dc6a8fd99be1de320e7169fef17c74ca2d3923053aa","last_reissued_at":"2026-05-17T23:57:01.950679Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:57:01.950679Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some properties of the distribution of the numbers of points on elliptic curves over a finite prime field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"James Mc Laughlin, Saiying He","submitted_at":"2019-01-03T03:59:30Z","abstract_excerpt":"Let $p \\geq 5$ be a prime and for $a, b \\in \\mathbb{F}_{p}$, let $E_{a,b}$ denote the elliptic curve over $\\mathbb{F}_{p}$ with equation $y^2=x^3+a\\,x + b$. As usual define the trace of Frobenius $a_{p,\\,a,\\,b}$ by \\begin{equation*}\n  \\#E_{a,b}(\\mathbb{F}_{p}) = p+1 -a_{p,\\,a,\\,b}. \\end{equation*} We use elementary facts about exponential sums and known results about binary quadratic forms over finite fields to evaluate the sums $\\sum_{t\\in\\mathbb{F}_{p}} a_{p,\\, t,\\, b}$, $\\sum _{t \\in \\mathbb{F}_{p}} a_{p,\\,a,\\, t}$,\n  $ \\sum_{t=0}^{p-1}a_{p,\\,t,\\,b}^{2}$, $ \\sum_{t=0}^{p-1}a_{p,\\,a,\\,t}^{2}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.00604","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":"1901.00604","created_at":"2026-05-17T23:57:01.950748+00:00"},{"alias_kind":"arxiv_version","alias_value":"1901.00604v1","created_at":"2026-05-17T23:57:01.950748+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.00604","created_at":"2026-05-17T23:57:01.950748+00:00"},{"alias_kind":"pith_short_12","alias_value":"JYSAJKHU6XFP","created_at":"2026-05-18T12:33:21.387695+00:00"},{"alias_kind":"pith_short_16","alias_value":"JYSAJKHU6XFPGXWD","created_at":"2026-05-18T12:33:21.387695+00:00"},{"alias_kind":"pith_short_8","alias_value":"JYSAJKHU","created_at":"2026-05-18T12:33:21.387695+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/JYSAJKHU6XFPGXWDHXDKR7MZXY","json":"https://pith.science/pith/JYSAJKHU6XFPGXWDHXDKR7MZXY.json","graph_json":"https://pith.science/api/pith-number/JYSAJKHU6XFPGXWDHXDKR7MZXY/graph.json","events_json":"https://pith.science/api/pith-number/JYSAJKHU6XFPGXWDHXDKR7MZXY/events.json","paper":"https://pith.science/paper/JYSAJKHU"},"agent_actions":{"view_html":"https://pith.science/pith/JYSAJKHU6XFPGXWDHXDKR7MZXY","download_json":"https://pith.science/pith/JYSAJKHU6XFPGXWDHXDKR7MZXY.json","view_paper":"https://pith.science/paper/JYSAJKHU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1901.00604&json=true","fetch_graph":"https://pith.science/api/pith-number/JYSAJKHU6XFPGXWDHXDKR7MZXY/graph.json","fetch_events":"https://pith.science/api/pith-number/JYSAJKHU6XFPGXWDHXDKR7MZXY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JYSAJKHU6XFPGXWDHXDKR7MZXY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JYSAJKHU6XFPGXWDHXDKR7MZXY/action/storage_attestation","attest_author":"https://pith.science/pith/JYSAJKHU6XFPGXWDHXDKR7MZXY/action/author_attestation","sign_citation":"https://pith.science/pith/JYSAJKHU6XFPGXWDHXDKR7MZXY/action/citation_signature","submit_replication":"https://pith.science/pith/JYSAJKHU6XFPGXWDHXDKR7MZXY/action/replication_record"}},"created_at":"2026-05-17T23:57:01.950748+00:00","updated_at":"2026-05-17T23:57:01.950748+00:00"}