curve #3709

y² = x³ + x² − 6760656887276461x + 214690902534289434588764

a-invariants: [0, 1, 0, -6760656887276461, 214690902534289434588764]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/6ℤ
conductor (N): 21175657519450980
discriminant (Δ): -135450033508050952180840901239018335036218132400
Faltings height: 7.9011 ☆ record for ℤ/6ℤ torsion, rank ≥ 6
naive height: 120.9701
primes of bad reduction: 2, 3, 5, 7, 13, 19, 23, 29, 59, 1979, 2621
regulator: 5428.8480436150140651930515375267752948462494879741
submitted by: Michael Rubinstein
submitted at: 2026-09-30 01:47:28 UTC
last updated: 2026-09-30 01:47:28 UTC

Witness: 6 independent points log in to add more points →

(-8452951, 520801709415)
(21464924, 281893116960)
(69337229, 281558010825)
(196273181/4, 265162695525/8)
(377806936/9, 1882827141650/27)
(495136876/9, 2598883241500/27)

Commentary

Specialization at t = -21608/11571 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 01:47:28 UTC · history