curve #3622

y² = x³ − x² − 11946911268550493608x + 15893653014210611705201601712

a-invariants: [0, -1, 0, -11946911268550493608, 15893653014210611705201601712]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ
conductor (N): 43316108595835552276361984400
discriminant (Δ): 3941805189066359449421657489636418187816716929856000000
Faltings height: 9.5943
naive height: 143.3946
primes of bad reduction: 2, 3, 5, 11, 43, 191, 199, 227, 263, 269, 421, 449, 1571
regulator: 8017894301.99912547259034226242970111631592730288133763248765
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:21:19 UTC
last updated: 2026-09-29 19:21:19 UTC

Witness: 11 independent points log in to add more points →

(1615407172, 28460446799400)
(1939993962, 4246541094950)
(2002794953, 160455647484)
(4177832092, 197237214939600)
(8428679196, 716932725701392)
(277975518892, 146546823407014800)
(7359598258/9, 2205269635129550/27)
(48131888314/25, 672549719058162/125)
(422519133212/961, 3085345472821894800/29791)
(1147353037924/529, 165399112248905304/12167)
(23799337064332/11881, 308083882692366000/1295029)

Commentary

Specialization at T = -40/9 of the rank-9 fibration of X400, y^2 + (Lx+B)y = x(x+D)(x+E), given in the commentary of curve #802 (same T), 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 19:21:19 UTC · history