curve #3716

y² + xy = x³ − 237906519865576420215x + 1426402656891806968184602251817

a-invariants: [1, 0, 0, -237906519865576420215, 1426402656891806968184602251817]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/6ℤ
conductor (N): 269191330439443485710
discriminant (Δ): -17172650899595984038207117793077656231964053574944845824000000
Faltings height: 10.5633 ☆ record for ℤ/6ℤ torsion, rank ≥ 7
naive height: 152.3886 ☆ record for ℤ/6ℤ torsion, rank ≥ 7
primes of bad reduction: 2, 5, 7, 13, 17, 43, 53, 79, 139, 569, 1222003
regulator: 56739.30484377825427389786365843319313576487286790025
submitted by: Michael Rubinstein
submitted at: 2026-09-30 01:48:51 UTC
last updated: 2026-09-30 01:48:51 UTC

Witness: 7 independent points log in to add more points →

(247802534, 1169386054826773)
(34132147574, 5750653636612213)
(-26674500933514/2209, 165416647633593451739/103823)
(-141992979044/9, 30220234866645061/27)
(50689523549/16, 53717587600938387/64)
(1258430074454/121, 369475346358705503/1331)
(84468876807686/4489, 571629878379476442519/300763)

Commentary

Specialization at t = -282883/156520 of the universal curve with a 6-torsion point, y^2 + txy + (t+2)y = x^3 (as searched in arXiv:2003.00077, Sec. 13), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-30 01:48:51 UTC · history