curve #3659

y² = x³ + 1745421357300x + 128967401282614000

a-invariants: [0, 0, 0, 1745421357300, 128967401282614000]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/4ℤ
conductor (N): 198468505684800
discriminant (Δ): -347500074979269068090778585999360000000
Faltings height: 6.0847
naive height: 96.1777
primes of bad reduction: 2, 3, 5, 13, 31, 151, 293, 773
regulator: 15830.856382367845382804284349234198066702910120403
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:28:28 UTC
last updated: 2026-09-29 19:28:28 UTC

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

(784830, 1407923600)
(38000990, 234398689200)
(77705534, 685079140752)
(4885316/49, 189110155536/343)
(12698446/9, 62745068656/27)
(119020145/1024, 18920761180425/32768)

Commentary

Specialization at t = -11/16 of the Elkies-Klagsbrun Z/4 fibration E1: y^2 + a*xy + a*c*y = x^3 + c*x^2, a = (8t-1)(32t+7), c = 8(t+1)(15t-8)(31t-7) (arXiv:2003.00077, eq. (5)), 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 19:28:28 UTC · history