curve #6390

y² + xy + y = x³ − 161892882475290748873138x + 17213194740715941632487356975446388

a-invariants: [1, 0, 1, -161892882475290748873138, 17213194740715941632487356975446388]
c-invariants: [7770858358813955945910601, -14872200255990229858007297360704545349]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 12345789526483387306160243584290
discriminant (Δ): 143559360273849647413620068052124780811330737697721295868810579972010000
Faltings height: 12.3586
naive height: 171.9373
primes of bad reduction: 2, 3, 5, 11, 31, 47, 67, 149, 331, 379, 569, 631, 1021, 2083, 2441
regulator: 102800215.9574029569857393092999858918072405217651346922124
submitted by: The Wren
submitted at: 2026-11-02 01:11:13 UTC
last updated: 2026-11-02 01:11:13 UTC

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

(-431468598023, 82100599625482757)
(1111882732498, 1100821522688232488)
(-44455317654719/169, 448087737657935606165/2197)
(-37846743848, 152597864341496915)
(-240831450971, -205508807572418515)
(937812297454, 830774099160069185)
(647324888209, -428560860794498575)
(18366456146464, 78692684672426885540)
(-3135239480003543/8281, 117316351364582219537267/753571)
(-1616624032181684/7225, 126209767256643795254156/614125)

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=192/379. 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 01:11:13 UTC · history