curve #5052

y² = x³ − 4887891348792611614545962404036x

a-invariants: [0, 0, 0, -4887891348792611614545962404036, 0]
c-invariants: [234618784742045357498206195393728, 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): 78206261580681785832735398464576
discriminant (Δ): 7473853912566332254632952684780655154177148553190345353200947212874875063531485923913907113984
Faltings height: 16.7021
naive height: 223.6065
primes of bad reduction: 2, 7, 17, 31, 41, 71, 1609, 63977
regulator: 2536671208.046682482901675556086640626215374186570069
submitted by: David Renshaw
submitted at: 2026-10-21 01:50:15 UTC
last updated: 2026-10-21 01:50:15 UTC

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

(5021263312917826434649/746496, 336042530229492048500034915277475/644972544)
(40249934287301444434/6561, 238198027584946084809727121360/531441)
(27399556163232609567058/10962721, 2115283342657849841858487498436848/36297569231)
(757322711808724936478674/3783025, 659014523635375563427373218432699968/7357983625)
(686026594953575253436153522/23694137041, 17916022537587333950941018695592337728464/3647214820584089)
(129859639632476651429077457/16699359076, 1418760872087327205473578355272370243079/2157991375955176)
(30334927397988297085228801/12063946896, 79591708516753007056380017355020763265/1325055671269056)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 2210857604820494, 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 01:50:15 UTC · history