curve #2666

y² + xy = x³ − 8397613456x + 252279426801920

a-invariants: [1, 0, 0, -8397613456, 252279426801920]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 9682530
discriminant (Δ): 10405980128230749229962263040000
Faltings height: 4.6742
naive height: 80.1672
primes of bad reduction: 2, 3, 5, 11, 13, 37, 61
regulator: 14.6059507234668463041337710942619962027766
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:16:45 UTC
last updated: 2026-09-26 20:16:45 UTC

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

(-89182, 17129516)
(-24832, 21119216)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-25, q=12; A=(3q-p)(p+q)^3=-134017, B=-(p+3q)(p-q)^3=557183. 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 20:16:45 UTC · history