curve #2659

y² + xy + y = x³ − 12469849453x − 510984467487244

a-invariants: [1, 0, 1, -12469849453, -510984467487244]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 8645010
discriminant (Δ): 11300707893288624291305123062500
Faltings height: 4.7155
naive height: 81.3533
primes of bad reduction: 2, 3, 5, 11, 17, 23, 67
regulator: 11.9662052506001572143583411612297068344079
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:16:11 UTC
last updated: 2026-09-26 20:16:11 UTC

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

(-74129, 2496224)
(-55175, 3039572)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-34, q=11; A=(3q-p)(p+q)^3=-815189, B=-(p+3q)(p-q)^3=-91125. 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 20:16:12 UTC · history