curve #2533

y² + xy + y = x³ − x² − 20320943x + 8757439031

a-invariants: [1, -1, 1, -20320943, 8757439031]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 18810
discriminant (Δ): 503953215403595858841600
Faltings height: 3.2390
naive height: 62.0951
primes of bad reduction: 2, 3, 5, 11, 19
regulator: 3.184275127414574742105207968286724541208
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:07:32 UTC
last updated: 2026-09-26 20:07:32 UTC

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

(-4635, 60232)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-15, q=4; A=(3q-p)(p+q)^3=-35937, B=-(p+3q)(p-q)^3=-20577. 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 20:07:32 UTC · history