curve #5561

y² + xy = x³ − 19392503243026609164490x + 1037895126441079298664944432605667

a-invariants: [1, 0, 0, -19392503243026609164490, 1037895126441079298664944432605667]
c-invariants: [930840155665277239895521, -896741389246488774280009905631139569]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 2234537902161497627835
discriminant (Δ): 1385302467423396321208134040142951812985763733732640726003000390625
Faltings height: 11.5857
naive height: 165.5711
primes of bad reduction: 3, 5, 7, 43, 61, 109, 113, 241, 1097, 1259, 1979
regulator: 1337719.538311317039914221618833478891998368123536898
submitted by: Daniel Błażewicz
submitted at: 2026-10-23 06:11:44 UTC
last updated: 2026-10-23 06:11:44 UTC

(-132020217826, 36014960716370813)
(41909782349, 17284976277474713)
(63264893399, 8015256395455613)
(85810476554, 2382245006347643)
(313411708589, 160453942093537433)
(-610926779840509/3844, 2448836480138553567409/238328)
(-7841081198299/49, 2260413048502642559/343)

(u,v) = (142267,987300) for the C2 x C4 family y^2 = x(x+u^2)(x+v^2). PARI/GP gives exact rank 7.

last edited by Daniel Błażewicz at 2026-10-23 06:11:44 UTC · history