curve #4721

y² = x³ − 43714839586459057956x

a-invariants: [0, 0, 0, -43714839586459057956, 0]
c-invariants: [2098312300150034781888, 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): 699437433383344927296 ☆ record for congruent-number curves, rank ≥ 6
discriminant (Δ): 5346463929781116455900283179857035792614088511271306289844224 ☆ record for congruent-number curves, rank ≥ 6
Faltings height: 10.3421 ☆ record for congruent-number curves, rank ≥ 6
naive height: 147.2863 ☆ record for congruent-number curves, rank ≥ 6
primes of bad reduction: 2, 3, 17, 31, 449, 4657
regulator: 2680449.3325836379883504750827325607287798840076862
submitted by: David Renshaw
submitted at: 2026-10-20 21:25:57 UTC
last updated: 2026-10-20 21:25:57 UTC

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

(165541824817/16, 51806810701954601/64)
(271329342673/16, 130148817129300583/64)
(-156630507, 82723846945707)
(-204975397538/81, 224003566698561944/729)
(548503784857/36, 365985935192610019/216)
(2724841474883505553/404492544, 861524446768349238596781127/8135154044928)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 6611719866. 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 21:25:57 UTC · history