curve #4719

y² = x³ − 1572516x

a-invariants: [0, 0, 0, -1572516, 0]
c-invariants: [75480768, 0]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
families: y² = x³ + ax (quartic twists); y² = x³ − n²x (congruent-number curves)
conductor (N): 25160256 ☆ record for congruent-number curves, rank ≥ 3
discriminant (Δ): 248865785384491782144 ☆ record for congruent-number curves, rank ≥ 3
Faltings height: 2.6031 ☆ record for congruent-number curves, rank ≥ 3
naive height: 54.4182 ☆ record for congruent-number curves, rank ≥ 3
primes of bad reduction: 2, 3, 11, 19
regulator: 22.688064423647360564945346115224478674797737
submitted by: David Renshaw
submitted at: 2026-10-20 21:25:49 UTC
last updated: 2026-10-20 21:25:49 UTC

Witness: 3 independent points log in to add more points →

(-912, 25992)
(1650, 43560)
(-528, 26136)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 1254. 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 21:25:49 UTC · history