curve #2429

y² + xy = x³ − 5503327764816x + 4969196172882950400

a-invariants: [1, 0, 0, -5503327764816, 4969196172882950400]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 10748010
discriminant (Δ): 4396435535331793706632504934400
Faltings height: 5.6982
naive height: 99.6227
primes of bad reduction: 2, 3, 5, 7, 13, 31, 127
regulator: 4.322980429480206310529624649049299933216
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:01:03 UTC
last updated: 2026-09-26 20:01:03 UTC

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

(1342248, 23819496)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-31, q=32; A=(3q-p)(p+q)^3=127, B=-(p+3q)(p-q)^3=16253055. 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 20:01:03 UTC · history