curve #2421

y² + xy + y = x³ − 33847953913x − 2309569542692344

a-invariants: [1, 0, 1, -33847953913, -2309569542692344]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 9699690
discriminant (Δ): 177529228623962620992500976562500
Faltings height: 4.9533
naive height: 84.3490
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19
regulator: 5.000094929790965463161921139808691686372
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:00:33 UTC
last updated: 2026-09-26 20:00:33 UTC

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

(-379925/4, 56085571/8)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-38, q=13; A=(3q-p)(p+q)^3=-1203125, B=-(p+3q)(p-q)^3=132651. 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 20:00:33 UTC · history