curve #3668

y² = x³ + x² − 40653066371981852305x + 99767215262970013788832370975

a-invariants: [0, 1, 0, -40653066371981852305, 99767215262970013788832370975]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/4ℤ
conductor (N): 6913619279132804573280
discriminant (Δ): 4492840643437444928902245022525569668498451000000000000
Faltings height: 9.8091
naive height: 147.0684
primes of bad reduction: 2, 3, 5, 11, 29, 41, 59, 67, 97, 137, 419, 50033
regulator: 2377942.42639535674411903786424790127261954615225766259
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:29:58 UTC
last updated: 2026-09-29 19:29:58 UTC

Witness: 8 independent points log in to add more points →

(3676736690, 406164886875)
(3678737315, 115276012500)
(3797207315, 12255776032500)
(40050327185/49, 88839868033702500/343)
(327994438315/121, 129640534871490300/1331)
(640171045115/169, 24777076709704740/2197)
(721083090365/196, 34242487459125/2744)
(106550621894915/32041, 210915819716901997500/5735339)

Commentary

Specialization at t = 137/54 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 19:29:58 UTC · history