curve #4731

y² = x³ − 45664130806096753844100x

a-invariants: [0, 0, 0, -45664130806096753844100, 0]
c-invariants: [2191878278692644184516800, 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): 730626092897548061505600
discriminant (Δ): 6094043647248539371477918240381094056566185517719091569350407744000000
Faltings height: 12.0799
naive height: 168.1404
primes of bad reduction: 2, 3, 5, 17, 103, 1249, 3257
regulator: 9537625.0794584170331012754066664824450602457304362
submitted by: David Renshaw
submitted at: 2026-10-20 21:26:38 UTC
last updated: 2026-10-20 21:26:38 UTC

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

(149669647271916/49, 1826565232243870153764/343)
(-135022380906, 60861159901278072)
(45030121627046640/157609, 6342661252892826238293000/62570773)
(1157482044084685/441, 39248875014125135685425/9261)
(2566032106948232640/11336689, 1355219411203866924163285200/38170631863)
(-1601233843245878531514/83037914569, 707157180081032939686674114839784/23928454575946747)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 213691672290. 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 21:26:38 UTC · history