curve #2409

y² + xy = x³ − x² − 256602376809x + 50029436179381113

a-invariants: [1, -1, 0, -256602376809, 50029436179381113]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 41840910
discriminant (Δ): 70083220425612460280551422562500
Faltings height: 5.1971
naive height: 90.4260
primes of bad reduction: 2, 3, 5, 17, 23, 29, 41
regulator: 3.686829237512784310701285986178863297770
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:59:49 UTC
last updated: 2026-09-26 19:59:49 UTC

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

(-355773, 310483824)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-18, q=23; A=(3q-p)(p+q)^3=10875, B=-(p+3q)(p-q)^3=3514971. 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 19:59:50 UTC · history