curve #4725

y² = x³ − 14213901280974016933921x

a-invariants: [0, 0, 0, -14213901280974016933921, 0]
c-invariants: [682267261486752812828208, 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): 454844840991168541885472
discriminant (Δ): 183789145461521004412264716742066272272186125689814156724395698493504
Faltings height: 11.7882
naive height: 164.6391
primes of bad reduction: 2, 3769, 4729, 6689
regulator: 496564848.84941929247945559856220755927643431822681
submitted by: David Renshaw
submitted at: 2026-10-20 21:26:13 UTC
last updated: 2026-10-20 21:26:13 UTC

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

(-225520462105552400/23319241, 1315922980592574487742406180/112608614789)
(-12537096713597551937600/34350019870321, 458543026396658098453521181462375560/201321653607745775369)
(253970200287659581174225/173652224656, 127563474012836868951204080308919655/72363660449749696)
(-265172088033772653963536/2374758050625, 51088291776632228802035470973705188/3659561524964390625)
(-61229933977612921062400/6920524614721, 203600278309385746546935850666810080/18205747978175772769)
(-4449186201994833087171326864/38819837726525025, 85023859702434342114580582724998796763116/7648575771762097995812625)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 119222067089. 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 21:26:13 UTC · history