curve #5020

y² = x³ − 44100x

a-invariants: [0, 0, 0, -44100, 0]
c-invariants: [2116800, 0]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
families: y² = x³ + ax (quartic twists); y² = x³ − n²x (congruent-number curves)
conductor (N): 705600
discriminant (Δ): 5489031744000000
Faltings height: 1.7096
naive height: 43.6962
primes of bad reduction: 2, 3, 5, 7
regulator: 2.91303967418348153861089862611940077594023
submitted by: David Renshaw
submitted at: 2026-10-21 01:47:37 UTC
last updated: 2026-10-21 01:47:37 UTC

Congruent-number curve y^2 = x^3 - n^2 x with n = 210. Found in an exhaustive ellrank scan of squarefree n <= 20000; one of the smallest n of rank 2.

last edited by David Renshaw at 2026-10-21 01:47:37 UTC · history