curve #6196

y² + xy + y = x³ − 209271536x + 1165774046610

a-invariants: [1, 0, 1, -209271536, 1165774046610]
c-invariants: [10045033705, -1007243843821813]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/3ℤ
conductor (N): 35858678834
discriminant (Δ): -560831625838385811301348
Faltings height: 3.5048
naive height: 69.0920
primes of bad reduction: 2, 11, 19, 37, 61, 191, 199
regulator: 633.628809154283738282911378738377566118736639233
submitted by: Andrii Prokhorenkov
submitted at: 2026-10-30 15:41:06 UTC
last updated: 2026-10-30 15:41:06 UTC

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

(2226, 842075)
(8881, 83405)
(247064, 122475070)
(865771/81, 278543197/729)
(2979771/361, 165150927/6859)

Commentary

Specialization at T = 81/191 of the elliptic fibration of X648 given by: y^2 + (Lx+B)y = x^3, where L = -91T^2 + 72T - 81, B = -81000T^5 + 243000T^4 - 243000T^3 + 81000T^2.

last edited by Andrii Prokhorenkov at 2026-10-30 15:41:06 UTC · history