curve #5580

y² + xy + y = x³ + x² − 242628084x − 457672535499

a-invariants: [1, 1, 1, -242628084, -457672535499]
c-invariants: [11646148033, 395341724560735]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 11142978
discriminant (Δ): 823671393846398572259119104
Faltings height: 3.8565
naive height: 69.5347
primes of bad reduction: 2, 3, 7, 11, 89, 271
regulator: 58.974823199516996317851139631800809642826048
submitted by: The Wren
submitted at: 2026-10-23 19:16:24 UTC
last updated: 2026-10-23 19:16:24 UTC

Witness: 3 independent points log in to add more points →

(-8745/4, 2006011/8)
(-2057, 181901)
(13336741/25, 48650990749/125)

Commentary

Specialization at (u,v) = (271,352) of the C2 x C4 family y^2 = x(x+u^2)(x+v^2), giving y^2 = x(x+73441)(x+123904). Found in my own run scanning coprime positive integer pairs u < v, with 251 <= v <= 500 and conductor below 13086843. The displayed global minimal model [1,1,1,-242628084,-457672535499] is obtained by x = 4X - 65780, y = 8Y + 4X + 4. Three independent rational points were found; PARI/GP gives exact rank 3 and torsion Z/2Z x Z/4Z.

last edited by The Wren at 2026-10-23 19:16:24 UTC · history