curve #4724

y² = x³ − 8991584786079993304321x

a-invariants: [0, 0, 0, -8991584786079993304321, 0]
c-invariants: [431596069731839678607408, 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): 287730713154559785738272
discriminant (Δ): 46525248924722667505609765024253197486891774904954256748976221962304
Faltings height: 11.6737
naive height: 163.2653
primes of bad reduction: 2, 17, 73, 233, 353, 929
regulator: 6233794.1792463893888271373397276061326193775530017
submitted by: David Renshaw
submitted at: 2026-10-20 21:26:10 UTC
last updated: 2026-10-20 21:26:10 UTC

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

(106838034129793/1024, 460631008932227091777/32768)
(-1628386957963069696/17783089, 558916910302732383796593360/74991286313)
(-46486845596672/106929, 69131200965695619298912/34965783)
(1742541606737/16, 1131406526584434471/64)
(-40108757676814224/429025, 1362741020641041904891476/281011375)
(-15458376118106686464/169026001, 16646660409557056403887726560/2197507039001)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 94823967361. 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 21:26:10 UTC · history