curve #3382

y² + xy + y = x³ − 1501515067102993x + 22394231806462694439056

a-invariants: [1, 0, 1, -1501515067102993, 22394231806462694439056]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 24657882990
discriminant (Δ): 6468810320606308281532898319099228057562500
Faltings height: 7.3435
naive height: 116.4494
primes of bad reduction: 2, 3, 5, 13, 37, 61, 109, 257
regulator: 153.32498789762503080526497485266912546580928
submitted by: David Renshaw
submitted at: 2026-09-26 21:18:05 UTC
last updated: 2026-09-26 21:18:05 UTC

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

(22577260, 1570757807)
(60985793905/256, 14872204020005327/4096)
(99747077365/4489, 326870085411866/300763)

Commentary

Found in a rational-parameter search of the Kubert–Tate family for torsion ℤ/2ℤ × ℤ/6ℤ: y^2 + (1-c)xy - by = x^3 - bx^2 with c=(10-2t)/(t^2-9), b=c+c^2, at t = 187/135; converted to the global minimal model. The 3 witness points were found with PARI/GP ellrank. Conductor 24657882990, torsion ℤ/2ℤ × ℤ/6ℤ.

last edited by David Renshaw at 2026-09-26 21:18:05 UTC · history