curve #2250

y² + xy + y = x³ − x² − 10869262320677x − 9841027541477543971

a-invariants: [1, -1, 1, -10869262320677, -9841027541477543971]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 81171090
discriminant (Δ): 40345239570953289900532186360274496000000
Faltings height: 6.4982
naive height: 101.6645
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 53
regulator: 2.698801584891250815199282217957522179165
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:49:46 UTC
last updated: 2026-09-26 19:49:46 UTC

Witness: 1 independent point log in to add more points →

(-1010613, 334288756)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-3, q=52; A=(3q-p)(p+q)^3=18706191, B=-(p+3q)(p-q)^3=25455375. The listed points are independent modulo torsion by the exact 2-descent Kummer squareclass map. The rank is a proven lower bound; no upper bound is claimed.

last edited by Natanael Wildner Fraga at 2026-09-26 19:49:46 UTC · history