curve #5003

y² = x³ − 169x

a-invariants: [0, 0, 0, -169, 0]
c-invariants: [8112, 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): 5408
discriminant (Δ): 308915776
Faltings height: 0.3185
naive height: 27.0033
primes of bad reduction: 2, 13
regulator: 5.832511736987650557951255312373859988603
submitted by: David Renshaw
submitted at: 2026-10-21 01:46:19 UTC
last updated: 2026-10-21 01:46:19 UTC

Congruent-number curve y^2 = x^3 - n^2 x with n = 13. 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 01:46:19 UTC · history