curve #4716

y² = x³ − 25x

a-invariants: [0, 0, 0, -25, 0]
c-invariants: [1200, 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): 800
discriminant (Δ): 1000000 ☆ record for congruent-number curves, rank ≥ 1
Faltings height: -0.1592 ☆ record for congruent-number curves, rank ≥ 1
naive height: 21.2702 ☆ record for congruent-number curves, rank ≥ 1
primes of bad reduction: 2, 5
regulator: 1.899482172531795590107205509594597919673
submitted by: David Renshaw
submitted at: 2026-10-20 21:25:40 UTC
last updated: 2026-10-20 21:25:40 UTC

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

(-4, 6)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 5. Source: Watkins, Donnelly, Elkies, Fisher, Granville, Rogers, 'Ranks of quadratic twists of elliptic curves', Publ. Math. Besancon 2014/2, 63-98, doi:10.5802/pmb.9 (https://pmb.centre-mersenne.org/item/10.5802/pmb.9.pdf), Table 2 'Smallest rank r quadratic twists y^2=x^3-d^2x'.

last edited by David Renshaw at 2026-10-20 21:25:40 UTC · history