Pith Number

pith:YHVAR3RE

pith:2026:YHVAR3REFM6XAGRYFIITHVPWSO

not attested not anchored not stored refs resolved

Kodaira-Neron statistics for rational elliptic curves with $j$-invariant 0 and 1728

John Cullinan, Sebastian Sargenti

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.

arxiv:2605.14226 v1 · 2026-05-14 · math.NT

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{YHVAR3REFM6XAGRYFIITHVPWSO}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest 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.

C2weakest 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.

C3one 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.

References

10 extracted · 10 resolved · 0 Pith anchors

[1] A. Barrios, M. Roy. Local data of rational elliptic curves with nontrivial torsion, Pacific J. Math.318no. 1, 1-42 (2022) 2022

[2] 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) 2022

[3] R. Brown. The natural density of some sets of square-free numbers. Integers 21 (2021), Paper No. A81, 9 pp 2021

[4] 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) 2021

[5] 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) 2022

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-17T23:39:10.778235Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

c1ea08ee242b3d701a382a1133d5f6939f7ff3ab3f9e1c282d1c82774378128c

Aliases

arxiv: 2605.14226 · arxiv_version: 2605.14226v1 · doi: 10.48550/arxiv.2605.14226 · pith_short_12: YHVAR3REFM6X · pith_short_16: YHVAR3REFM6XAGRY · pith_short_8: YHVAR3RE

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/YHVAR3REFM6XAGRYFIITHVPWSO \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: c1ea08ee242b3d701a382a1133d5f6939f7ff3ab3f9e1c282d1c82774378128c

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "cb6028f4c1fb78ec9a09a4a6811d41e8798d885c7ae5a4b026837eaacd7bb503",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-05-14T00:44:44Z",
    "title_canon_sha256": "97828ec6f9b54256a1bc7e146f3eb92b3745331956b9ad7e4d7c2d8961828cca"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.14226",
    "kind": "arxiv",
    "version": 1
  }
}