curve #4723

y² = x³ − 4440543488446744649476x

a-invariants: [0, 0, 0, -4440543488446744649476, 0]
c-invariants: [213146087445443743174848, 0]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
families: y² = x³ + ax (quartic twists); y² = x³ − n²x (congruent-number curves)
conductor (N): 71048695815147914391616
discriminant (Δ): 5603873937673552989300498033450919374305569702372133792442324619264
Faltings height: 11.4973
naive height: 161.1488
primes of bad reduction: 2, 11, 41, 73, 83, 89, 137
regulator: 3563463.7864788079532444667284295752395634886940862
submitted by: David Renshaw
submitted at: 2026-10-20 21:26:05 UTC
last updated: 2026-10-20 21:26:05 UTC

Witness: 6 independent points log in to add more points →

(9545443187049/100, 21115589873934251157/1000)
(73145926991313/784, 437846569428939872487/21952)
(-107791645753616/2025, 842818572202479027848/91125)
(443592838138329/2704, 8537296635485843022435/140608)
(4279322629593/64, 728847146421825123/512)
(22114530754212226/140625, 2978793707011327140698224/52734375)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 66637403074. Source: Watkins et al. electronic data http://magma.maths.usyd.edu.au/~watkins/PTS.r6r7 (paper Sec. 15); 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).

last edited by David Renshaw at 2026-10-20 21:26:06 UTC · history