curve #2656

y² + xy + y = x³ − 17796678684x + 886861020485146

a-invariants: [1, 0, 1, -17796678684, 886861020485146]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 570570
discriminant (Δ): 20963275841017002738621185409600
Faltings height: 4.7825
naive height: 82.4204
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19
regulator: 5.00170973675726011649343038055202521505192
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:15:57 UTC
last updated: 2026-09-26 20:15:57 UTC

Witness: 2 independent points log in to add more points →

(-152452, 7611198)
(-142762, 22829343)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-8, q=19; A=(3q-p)(p+q)^3=86515, B=-(p+3q)(p-q)^3=964467. 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 20:15:57 UTC · history