curve #2499

y² + xy = x³ − 896330415x + 10327643775225

a-invariants: [1, 0, 0, -896330415, 10327643775225]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 6557070
discriminant (Δ): 9745766524619334729000000
Faltings height: 3.8186
naive height: 73.4551
primes of bad reduction: 2, 3, 5, 13, 17, 23, 43
regulator: 4.115841219030669213246965211142917051589
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:05:23 UTC
last updated: 2026-09-26 20:05:23 UTC

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

(-133197/4, 14558751/8)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-13, q=10; A=(3q-p)(p+q)^3=-1161, B=-(p+3q)(p-q)^3=206839. 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 20:05:24 UTC · history