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We study the family of all such curves and also several natural subfamilies, including those with fixed $ j $-invariant and those with complex multiplication (CM). In particular, we provide formulas for two commonly used normalizations of the naive height appearing in the literature: the calibrated naive height, defined by \\[ H^{\\mathrm{cal}}(E_{A,B}) := \\max\\{ 4|A|^3, 27B^2 \\}, \\] and the uncalibrated naive height, defin"},"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":"2506.18874","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-06-23T17:43:38Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"67ea42747f8dc3e76df29f247ce22a32994e4af7818f1418dfd24d15be280f20","abstract_canon_sha256":"0dfe72997cf3724d8a1db15e6efa2eb1f109b6e5d8a448ea53335570a35e48f9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-08T01:03:45.901931Z","signature_b64":"+YEiHrnaVO32uFMi9gEXW2d4oMzi9qJQ87w3RO4GDEozgDUbokc16wkpSOGrJNVqOsf+zaza3u2bJ27TkJQMBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ddcafa7cf5abe1bc1494f14c8d939ec47d9a8c621ed93787a7a1966acfe8a175","last_reissued_at":"2026-06-08T01:03:45.900963Z","signature_status":"signed_v1","first_computed_at":"2026-06-08T01:03:45.900963Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Counting elliptic curves over $\\mathbb{Q}$ with bounded naive height","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Adrian Barquero-Sanchez, Daniel Mora-Mora","submitted_at":"2025-06-23T17:43:38Z","abstract_excerpt":"In this paper, we give exact and asymptotic formulas for counting elliptic curves $ E_{A,B} \\colon y^2 = x^3 + Ax + B $ with $ A, B \\in \\mathbb{Z} $, ordered by naive height. We study the family of all such curves and also several natural subfamilies, including those with fixed $ j $-invariant and those with complex multiplication (CM). 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