curve #2389

y² + xy + y = x³ − 82483780684x + 1543480349567546

a-invariants: [1, 0, 1, -82483780684, 1543480349567546]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 27999510
discriminant (Δ): 34886632350765265330922544382592400
Faltings height: 5.3182
naive height: 87.0212
primes of bad reduction: 2, 3, 5, 7, 11, 17, 23, 31
regulator: 3.570057421099434632127832332576234965564
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:58:31 UTC
last updated: 2026-09-26 19:58:31 UTC

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

(-92744, 91674017)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-4, q=27; A=(3q-p)(p+q)^3=1034195, B=-(p+3q)(p-q)^3=2293907. 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 19:58:31 UTC · history