curve #4718

y² = x³ − 1156x

a-invariants: [0, 0, 0, -1156, 0]
c-invariants: [55488, 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): 18496 ☆ record for congruent-number curves, rank ≥ 2
discriminant (Δ): 98867482624 ☆ record for congruent-number curves, rank ≥ 2
Faltings height: 0.7992 ☆ record for congruent-number curves, rank ≥ 2
naive height: 32.7718 ☆ record for congruent-number curves, rank ≥ 2
primes of bad reduction: 2, 17
regulator: 7.09967518240335471222269992268930159230274
submitted by: David Renshaw
submitted at: 2026-10-20 21:25:44 UTC
last updated: 2026-10-20 21:25:44 UTC

Witness: 2 independent points log in to add more points →

(-2, 48)
(-16, 120)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 34. 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 21:25:44 UTC · history