curve #6198

y² + xy = x³ − 21348x + 1198864

a-invariants: [1, 0, 0, -21348, 1198864]
c-invariants: [1024705, -1037355553]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/3ℤ
conductor (N): 1137742
discriminant (Δ): -84174704128
Faltings height: 1.1435
naive height: 41.5199
primes of bad reduction: 2, 17, 109, 307
regulator: 9.3245047429259129515052711629263544347115050
submitted by: Andrii Prokhorenkov
submitted at: 2026-10-30 15:41:29 UTC
last updated: 2026-10-30 15:41:29 UTC

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

(-124, 1456)
(46, 538)
(284, 4108)

Commentary

Specialization at T = -3/17 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 15:41:29 UTC · history