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The first coefficient is zero if and only if the division polynomial has no roots, which is equivalent to E being supersingular. Deuring (1941) proved that this supersingularity is also equivalent to the vanishing of the second coefficient. So the zero loci of the coefficients (as functions of A and B) are equal; the main resu"},"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":"1303.5002","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-03-20T17:38:38Z","cross_cats_sorted":[],"title_canon_sha256":"28c629d9d96aa023caca335d74f78c5f4bd492539029abc98156fb4c391e8963","abstract_canon_sha256":"61a9d517894b32477a513ab46acb5929150094b84991c47c48dd649aa3605be3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:30:15.441047Z","signature_b64":"uBFzlUyS5QDP+GJEU0lO1dabQX7lnu7zvsJf4y69qItVPA7xCDmQV/8mS92qYoL9/ItO9V9knX2/9Q8OMwI8Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"33a6321464c8eeb6685436e267add33cecdfd490cbdefc3237bffd10fec63308","last_reissued_at":"2026-05-18T03:30:15.439964Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:30:15.439964Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Beyond two criteria for supersingularity: coefficients of division polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Christophe Debry","submitted_at":"2013-03-20T17:38:38Z","abstract_excerpt":"Let E: y^2 = x^3 + Ax + B be an elliptic curve defined over a finite field of characteristic p\\geq 3. 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