curve #4727

y² = x³ − 16756952554205114492241x

a-invariants: [0, 0, 0, -16756952554205114492241, 0]
c-invariants: [804333722601845495627568, 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): 536222481734563663751712
discriminant (Δ): 301137675666469835854802414497033408718907851837883666656813795425344
Faltings height: 11.8293
naive height: 165.1329
primes of bad reduction: 2, 3, 7, 11, 13, 37, 61, 71, 269
regulator: 1479717.7595592296353982910608390093202624199254129
submitted by: David Renshaw
submitted at: 2026-10-20 21:26:22 UTC
last updated: 2026-10-20 21:26:22 UTC

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

(-1144374670439/16, 1846740346294288445/64)
(12463643091900, 43999129162103505390)
(-82686378852, 28639865139644682)
(-92901152548976772/1324801, 43936729597125810198655902/1524845951)
(-875516686641168/16129, 56084494263053852586276/2048383)
(288728101140297/1936, 2436279053497332857541/85184)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 129448648329. 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 21:26:22 UTC · history