curve #5037

y² = x³ − 109482236161x

a-invariants: [0, 0, 0, -109482236161, 0]
c-invariants: [5255147335728, 0]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
families: y² = x³ + ax (quartic twists); y² = x³ − n²x (congruent-number curves)
conductor (N): 3503431557152
discriminant (Δ): 83986784002698650999120402575441984
Faltings height: 5.3908
naive height: 87.8707
primes of bad reduction: 2, 409, 809
regulator: 33406.65650862437273568790922879489194135651294
submitted by: David Renshaw
submitted at: 2026-10-21 01:49:07 UTC
last updated: 2026-10-21 01:49:07 UTC

(-263169, 102887280)
(-3681, 20073720)
(-370765741640239758400/1173417425342761, 2218938482319979433200266368520/40195626293670265094741)
(-86544400/289, 378359511420/4913)

Congruent-number curve y^2 = x^3 - n^2 x with n = 330881. Found in an ellrank scan of squarefree n <= 6*10^5 with n = 1,2,3 mod 8.

last edited by David Renshaw at 2026-10-21 01:49:07 UTC · history