curve #5054

y² = x³ − 139365242872766860931630162064100x

a-invariants: [0, 0, 0, -139365242872766860931630162064100, 0]
c-invariants: [6689531657892809324718247779076800, 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): 2229843885964269774906082593025600
discriminant (Δ): 173238096009020285000252496373565722232373011364077623604839208447358841233505580751682222144000000
Faltings height: 17.5397
naive height: 233.6576
primes of bad reduction: 2, 5, 11, 17, 43, 193, 409, 1193, 1559
regulator: 1456072155.091865677573843379580969752708624418007809
submitted by: David Renshaw
submitted at: 2026-10-21 01:50:23 UTC
last updated: 2026-10-21 01:50:23 UTC

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

(3551109721579398321765/219961, 144347200380223074305917559445075/103161709)
(75398094273297634565/6241, 139090003900240897824954102525/493039)
(4120565424257133841/324, 3111178132571593217692998439/5832)
(132426550038117209106529/583696, 48125345606832471814431676379324433/445943744)
(387097181553305641/4, 239042071682596306748359989/8)
(36396443980054413043569/4900, 6943650973367825308600005891643497/343000)
(613176294234149213628622969/49260914704, 4814119004880986781631038196865693098803/10933361496723392)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 11805305708568790, 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 01:50:23 UTC · history