curve #5009

y² = x³ − 841x

a-invariants: [0, 0, 0, -841, 0]
c-invariants: [40368, 0]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
families: y² = x³ + ax (quartic twists); y² = x³ − n²x (congruent-number curves)
conductor (N): 26912
discriminant (Δ): 38068692544
Faltings height: 0.7197
naive height: 31.8174
primes of bad reduction: 2, 29
regulator: 8.843712730699025408533339022402093288511
submitted by: David Renshaw
submitted at: 2026-10-21 01:46:43 UTC
last updated: 2026-10-21 01:46:43 UTC

Congruent-number curve y^2 = x^3 - n^2 x with n = 29. Found in an exhaustive ellrank scan of squarefree n <= 20000; one of the smallest n of rank 1.

last edited by David Renshaw at 2026-10-21 01:46:43 UTC · history