curve #3656

y² = x³ + x² − 1304100895809x + 573209971740588735

a-invariants: [0, 1, 0, -1304100895809, 573209971740588735]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/4ℤ
conductor (N): 50691232704
discriminant (Δ): 547233200555888196890935054958592
Faltings height: 5.5271
naive height: 95.3032
primes of bad reduction: 2, 3, 7, 19, 71, 73, 383
regulator: 449.806296513330274238482436861770554682269689399
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:27:58 UTC
last updated: 2026-09-29 19:27:58 UTC

Witness: 5 independent points log in to add more points →

(569327, 123645648)
(634434, 34759683)
(2645301/4, 20975601/8)
(31829241/64, 111762495789/512)
(370004223/529, 693383647344/12167)

Commentary

Specialization at t = -17/16 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 19:27:58 UTC · history