curve #4729

y² = x³ − 32214206051402031362601x

a-invariants: [0, 0, 0, -32214206051402031362601, 0]
c-invariants: [1546281890467297505404848, 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): 1030854593644865003603232
discriminant (Δ): 2139549165113804326253761711506788262837752357002901686727840407539264
Faltings height: 11.9927
naive height: 167.0937
primes of bad reduction: 2, 3, 7, 23, 41, 47, 79, 2441
regulator: 5570514.5063738469453365023128372304021285868883089
submitted by: David Renshaw
submitted at: 2026-10-20 21:26:30 UTC
last updated: 2026-10-20 21:26:30 UTC

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

(180351302908, 7505950700076598)
(-55602865769759/400, 338653441212364798289/8000)
(34467728131900/9, 202135003390522396810/27)
(1300206430317942364/4223025, 1204603703661215826536940638/8678316375)
(-8411121412463/64, 22688786300873973049/512)
(5473196888023973988817/30160574224, 59732630606441533167527846372711/5237926604333632)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 179483163699. 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 21:26:30 UTC · history