curve #5517

y² = x³ − x² − 4139970728655939220x + 3237272302994874661410568900

a-invariants: [0, -1, 0, -4139970728655939220, 3237272302994874661410568900]
c-invariants: [198718594975485082576, -2797003268595260137605821034176]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 160175046391510920
discriminant (Δ): 13881485973490995458663648341006776289405836524100000000
Faltings height: 9.4738
naive height: 140.2153
primes of bad reduction: 2, 3, 5, 7, 11, 31, 61, 71, 139, 367, 2531
regulator: 369013.61920028175925681024946707623397636484678305
submitted by: Daniel Błażewicz
submitted at: 2026-10-22 21:22:21 UTC
last updated: 2026-10-22 21:22:21 UTC

(324302896, 43917826788990)
(1245283397, 3596488743582)
(18984173860, 2601253142405190)
(55193534740, 12958096035115350)
(40021947579040/44521, 146166971950767670650/9393931)
(1265750898354090/1413721, 26489096451230415695000/1680914269)

(u,v) = (8662,59675) for the C2 x C4 family y^2 = x(x+u^2)(x+v^2). PARI/GP gives exact rank 6.

last edited by Daniel Błażewicz at 2026-10-22 21:22:21 UTC · history