curve #3710

y² = x³ + x² − 108273200568257236x + 13712908201854908926067204

a-invariants: [0, 1, 0, -108273200568257236, 13712908201854908926067204]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/6ℤ
conductor (N): 21175657519450980
discriminant (Δ): 2192346096586826707645695580418704252987994880 ☆ record for ℤ/6ℤ torsion, rank ≥ 6
Faltings height: 8.2476
naive height: 129.2839
primes of bad reduction: 2, 3, 5, 7, 13, 19, 23, 29, 59, 1979, 2621
regulator: 21715.392174460056260772206150107101179384997951897
submitted by: Michael Rubinstein
submitted at: 2026-09-30 01:47:40 UTC
last updated: 2026-09-30 01:47:40 UTC

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

(165107612, 580601914662)
(189405620, 13614868722)
(189753092, 5314797762)
(189864071, 2648833320)
(4739594404/25, 1170278148858/125)
(177737440856/625, 38018992336662246/15625)

Commentary

Specialization at t = -47818/767 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 01:47:40 UTC · history