curve #3986

y² + xy = x³ + x² − 138867859595759582x + 1227166924071872322092580

a-invariants: [1, 1, 0, -138867859595759582, 1227166924071872322092580]
c-invariants: [6665657260596459961, -1060272272390527140761438765]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ
conductor (N): 672519215577201888821790774558
discriminant (Δ): 170739324019474751329036099245369612564375238837732752
Faltings height: 8.9039
naive height: 130.0305
primes of bad reduction: 2, 3, 11, 41, 53, 61, 71, 89, 103, 191, 499, 1013, 5653
regulator: 56274266480667.2845464137010321755341309821676230521866278494
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:39:53 UTC
last updated: 2026-10-05 04:39:53 UTC

Witness: 11 independent points log in to add more points →

(-195710674916696/528529, 526282792526149196150/384240583)
(-778453226219/2809, 639115910449271499/148877)
(4210956285852136/41209, 273254632039838121956634/8365427)
(-25284927119411/73441, 57091307881770777444/19902511)
(-17336730037533791/157552704, 7704001034940032666506627/1977601540608)
(86520551464/169, -17619901521977962/2197)
(-433820928731/1369, -185490671708750853/50653)
(43418873604934024/82609921, -6434279369442195122134562/750841571969)
(683398000022790676/1076430481, -459071868233538669559628074/35316607651129)
(3206945868456/3481, 5257018378097333646/205379)
(11851844992423621/8497225, -1243682238468231905135964/24769410875)

Commentary

Unlisted rational-isogeny neighbor of curve#3624 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 11: 11 independent points certified by the unchanged ICARM verifier, with a matching unconditional upper bound computed using PARI/GP 2.17.2 ellrank. This is an additional representative of a known isogeny class, not a claim of a new world rank record.

last edited by Odbayar Shinebayar at 2026-10-05 04:39:53 UTC · history