curve #3384

y² + xy = x³ + 41629871161444x − 116836401524714977008

a-invariants: [1, 0, 0, 41629871161444, -116836401524714977008]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/7ℤ
conductor (N): 12751938654
discriminant (Δ): -10514497359084633769917825294479040209485824
Faltings height: 6.9410
naive height: 105.9378
primes of bad reduction: 2, 3, 7, 13, 19, 41, 4283
regulator: 285.05987295811494860156595926879788971885726
submitted by: David Renshaw
submitted at: 2026-09-26 21:18:26 UTC
last updated: 2026-09-26 21:18:26 UTC

Witness: 3 independent points log in to add more points →

(21888952, 106205668132)
(400026664/9, 8078788794172/27)
(15815721304/3025, 2597029580767132/166375)

Commentary

Found in a rational-parameter search of the Kubert–Tate family for torsion ℤ/7ℤ: y^2 + (1-c)xy - by = x^3 - bx^2 with b=t^3-t^2, c=t^2-t, at t = 104/123; converted to the global minimal model. The 3 witness points were found with PARI/GP ellrank. Conductor 12751938654, torsion ℤ/7ℤ.

last edited by David Renshaw at 2026-09-26 21:18:26 UTC · history