curve #5058

y² = x³ − 2164260278133322682221505212788676x

a-invariants: [0, 0, 0, -2164260278133322682221505212788676, 0]
c-invariants: [103884493350399488746632250213856448, 0]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
families: y² = x³ + ax (quartic twists); y² = x³ − n²x (congruent-number curves)
conductor (N): 34628164450133162915544083404618816
discriminant (Δ): 648796412806690626353528480103581576354921370438993521336666615049859843606598600913106984945163505664
Faltings height: 18.2254
naive height: 241.8858
primes of bad reduction: 2, 7, 23, 41, 73, 79, 193, 1297, 2441
regulator: 1930965131.396565564876709092621728318000168197247271
submitted by: David Renshaw
submitted at: 2026-10-21 01:50:38 UTC
last updated: 2026-10-21 01:50:38 UTC

Witness: 7 independent points log in to add more points →

(467885694620490752, 318458030287945869750849216)
(-115013181850350992/9, 136532037582074453462746408/27)
(-10779212024424390123809118/283686649, 25000474387539043229812326098929573344/4778134229107)
(-26096018576302086561170192/2634460929, 611728358246959024244811382942260631672/135218976102783)
(-1024153500246870387796185512862/67312359856921, 2994794101465844979067527390125953030127995392/552258265646081078531)
(-9466623698437582, 4431689018989664882695008)
(-1148973251350491248192/29929, 26657526734467803627821948369328/5177717)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 46521610872080974, rank 7 example from Watkins, Donnelly, Elkies, Fisher, Granville, Rogers, 'Ranks of quadratic twists of elliptic curves', Publ. Math. Besancon 2014, doi:10.5802/pmb.9; points from https://magma.maths.usyd.edu.au/~watkins/PTS.r6r7.

last edited by David Renshaw at 2026-10-21 01:50:38 UTC · history