{"paper":{"title":"Kodaira-Neron statistics for rational elliptic curves with $j$-invariant 0 and 1728","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Elliptic curves with j-invariant 0 or 1728 admit explicit asymptotic counts of their Kodaira-Néron reduction types at 3 and 2 when ordered by height.","cross_cats":[],"primary_cat":"math.NT","authors_text":"John Cullinan, Sebastian Sargenti","submitted_at":"2026-05-14T00:44:44Z","abstract_excerpt":"Elliptic curves over $\\Q$ with $j$-invariant 0 or 1728 have additive reduction at all primes of bad reduction. In addition, all elliptic curves with $j$-invariant 0 have bad reduction at $p=3$ and all elliptic curves with $j$-invariant 1728 have bad reduction at $p=2$. In this paper we count elliptic curves with $j$-invariant 0 and 1728 by height and determine asymptotics for the various Kodaira-N\\'eron types at 3 and 2, respectively. We also give related statistics by holding the torsion subgoup and isogeny-torsion graph constant."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We count elliptic curves with j-invariant 0 and 1728 by height and determine asymptotics for the various Kodaira-Néron types at 3 and 2, respectively. We also give related statistics by holding the torsion subgroup and isogeny-torsion graph constant.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the elliptic curves with these j-invariants are sufficiently dense or uniformly distributed by height to allow for asymptotic counts of reduction types.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Asymptotics for Kodaira-Néron types of j=0 and j=1728 elliptic curves over Q are determined, including when torsion subgroup is fixed.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Elliptic curves with j-invariant 0 or 1728 admit explicit asymptotic counts of their Kodaira-Néron reduction types at 3 and 2 when ordered by height.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"ef46cadc2307803e10978977a8726c439e13d0c86396b9505f2365012700353f"},"source":{"id":"2605.14226","kind":"arxiv","version":1},"verdict":{"id":"c6d65832-884e-4697-a16c-2859c7cd5609","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:45:30.838978Z","strongest_claim":"We count elliptic curves with j-invariant 0 and 1728 by height and determine asymptotics for the various Kodaira-Néron types at 3 and 2, respectively. We also give related statistics by holding the torsion subgroup and isogeny-torsion graph constant.","one_line_summary":"Asymptotics for Kodaira-Néron types of j=0 and j=1728 elliptic curves over Q are determined, including when torsion subgroup is fixed.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the elliptic curves with these j-invariants are sufficiently dense or uniformly distributed by height to allow for asymptotic counts of reduction types.","pith_extraction_headline":"Elliptic curves with j-invariant 0 or 1728 admit explicit asymptotic counts of their Kodaira-Néron reduction types at 3 and 2 when ordered by height."},"references":{"count":10,"sample":[{"doi":"","year":2022,"title":"A. Barrios, M. Roy. Local data of rational elliptic curves with nontrivial torsion, Pacific J. Math.318no. 1, 1-42 (2022)","work_id":"5736d1cc-06be-4d59-810e-c172782db4f3","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2022,"title":"A. Barrios, M. Roy. Representations attached to elliptic curves with a non-trivial odd torsion point, Bull. Lond. Math. Soc.54no. 5, 1846-1861 (2022)","work_id":"6e68850c-207a-4d93-9d47-0c9fc25d37af","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"R. Brown. The natural density of some sets of square-free numbers. Integers 21 (2021), Paper No. A81, 9 pp","work_id":"3e3929f2-72a9-4df0-b169-471e3c044529","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"G. Chiloyan, A. Lozano-Robledo. A classification of isogeny-torsion graphs ofQ-sogeny classes of elliptic curves. Trans. London Math. Soc.8no. 1, 1–34 (2021)","work_id":"1464455d-e657-4b99-9909-37c18e5399ab","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2022,"title":"J. Cullinan, M. Kenney, J. Voight. On a probabilistic local-global principle for torsion on elliptic curves. J. Th´ eorie Nombres Bordeaux.34(1) 41-90 (2022)","work_id":"bcefe842-3a9a-40ec-9a7b-864c744b1f12","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":10,"snapshot_sha256":"b171e35e4d59b9c46110723bd938000b879a8d8323df56983f6ce6a8ba0154be","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"5688001636005fd6bd0fb7c06e8b57fe56d89d92d5330bb0597b51fc193a9a27"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}