curve #3708

y² + xy + y = x³ − 856942416122994871x + 305334139242515029665195210

a-invariants: [1, 0, 1, -856942416122994871, 305334139242515029665195210]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/6ℤ
conductor (N): 1469940702224254 ☆ record for ℤ/6ℤ torsion, rank ≥ 6
discriminant (Δ): 38528381200959072443738103528525883999891989498512
Faltings height: 8.8422
naive height: 135.4900
primes of bad reduction: 2, 13, 19, 23, 29, 31, 37, 97, 101, 397
regulator: 13903.957632214866783653550860826586375289850738196
submitted by: Michael Rubinstein
submitted at: 2026-09-30 01:47:17 UTC
last updated: 2026-09-30 01:47:17 UTC

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

(-903695570, 18486447750880)
(-6255741, 17626537223820)
(530846676, 143750098770)
(533274211, 45632400208)
(541376286, 277006976433)
(154366989276/289, 18471216875145/4913)

Commentary

Specialization at t = 80479/2231 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 01:47:17 UTC · history