curve #3664

y² = x³ − x² − 750826076561680x + 11332358425729416579700

a-invariants: [0, -1, 0, -750826076561680, 11332358425729416579700]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/4ℤ
conductor (N): 1578385609386038880
discriminant (Δ): -28389137094151847682746819617320466960880524800
Faltings height: 7.6156
naive height: 115.0870
primes of bad reduction: 2, 3, 5, 7, 19, 31, 71, 101, 157, 281, 2521
regulator: 11416408.88265326190991398827303916597027007622670787
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:29:18 UTC
last updated: 2026-09-29 19:29:18 UTC

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

(119492417, 1275844565658)
(-2765676724470495/287675521, 648449096088957876959870/4879264511681)
(-166805600711/7744, 90179158020909645/681472)
(-19034700983/576, 103232596200283/13824)
(-330034023/16, 8596833630347/64)
(19180783677/961, 1952982748805552/29791)
(25811558356/225, 4042842304672054/3375)

Commentary

Specialization at t = -140 of the Elkies-Klagsbrun Z/4 fibration E2: y^2 + a*xy + a*c*y = x^3 + c*x^2, a = -8(80t+9), c = 2(t-2)(2t-1)(18t-1)(2t-81) (arXiv:2003.00077, eq. (6)), 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 2026-09-29 19:29:18 UTC · history