curve #6401

y² = x³ − 182972826267525396285876x + 13972928679784991179787881000739040

a-invariants: [0, 0, 0, -182972826267525396285876, 13972928679784991179787881000739040]
c-invariants: [8782695660841219021722048, -12072610379334232379336729184638530560]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 454308257228250524769691706722752
discriminant (Δ): 307703607982562298104841560119769919173006308028403545840148291256356864
Faltings height: 12.4124
naive height: 172.3045
primes of bad reduction: 2, 3, 11, 13, 17, 37, 59, 79, 109, 127, 379, 431, 433, 569, 8297
regulator: 202468013.6214289513814171659961108068892791566795767181371
submitted by: The Wren
submitted at: 2026-11-02 01:36:59 UTC
last updated: 2026-11-02 01:36:59 UTC

Witness: 10 independent points log in to add more points →

(-183264349422, -203347586342439792)
(74420729812, 27715204022909084)
(3225718091629, 5743533807012616615)
(-19649929895444838/96721, 6220175835797496727582824/30080231)
(390195010460310/529, 6441835998473046087840/12167)
(-5535434614743/16, 12120527885028796629/64)
(799266530181, 615078776590120485)
(-36943204512962/289, 922724875122407633672/4913)
(76502688702, -20561124542283864)
(-218337266763, 208600873037192691)

Commentary

Z/2xZ/2 X400a family y^2=x(x+A)(x+B), A=q(121q-99p)(26p-23q)(75p-44q), B=p(21q-74p)(25p-36q)(p+77q), at T=p/q=242/629. Family equations as posted by wgxli with curve #4252; section formulas from the Layr Labs Elliptic Rank Challenge repository (submission/research/x400a-family.gp, commit 7482bf9). T selected by our Mestre-Nagao sieve; PARI/GP 2.19 ellrank seeded with the family sections, bounds [10,10]; torsion Z/2xZ/2 (elltors).

last edited by The Wren at 2026-11-02 01:36:59 UTC · history