curve #1189

y² + xy + y = x³ + x² − 78912468240x + 8490058018974705

a-invariants: [1, 1, 1, -78912468240, 8490058018974705]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/8ℤ
conductor (N): 779644110
discriminant (Δ): 310455332599837597122176100000000 ☆ record for ℤ/8ℤ torsion, rank ≥ 3
Faltings height: 5.0761 ☆ record for ℤ/8ℤ torsion, rank ≥ 3
naive height: 86.8884 ☆ record for ℤ/8ℤ torsion, rank ≥ 3
primes of bad reduction: 2, 3, 5, 7, 23, 31, 41, 127
regulator: 72.940269891871439152821852606970462968352806
submitted by: David Renshaw
submitted at: 2026-09-23 21:40:11 UTC
last updated: 2026-09-23 21:40:11 UTC

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

(1553263/36, 15519728113/216)
(7438237/49, 1010404499/343)
(779400377/5329, 3389420987481/389017)

Commentary

Found by a rational-parameter search of the Kubert-Tate family for torsion 8, at t=-10/21. In y^2+(1-c)xy-by=x^3-bx^2 the parameters are defined by b=(2*t-1)*(t-1);c=b/t; Converted to the global minimal model; PARI/GP ellrank returned bounds [3,3]. The 3 supplied points and the torsion subgroup are independently certified by the site.

last edited by David Renshaw at 2026-09-23 21:40:11 UTC · history