curve #2331

y² + xy + y = x³ − 2379809715669x + 1412997643545194476

a-invariants: [1, 0, 1, -2379809715669, 1412997643545194476]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 235409370
discriminant (Δ): 79304725416241587760908583540140900
Faltings height: 5.7642
naive height: 97.1077
primes of bad reduction: 2, 3, 5, 7, 17, 23, 47, 61
regulator: 3.595489606864488820550129258795552656023
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:54:55 UTC
last updated: 2026-09-26 19:54:55 UTC

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

(788420, 163297272)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-34, q=27; A=(3q-p)(p+q)^3=-39445, B=-(p+3q)(p-q)^3=10668107. 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 19:54:55 UTC · history