curve #5050

y² = x³ − 319229978799412836x

a-invariants: [0, 0, 0, -319229978799412836, 0]
c-invariants: [15323038982371816128, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
families: y² = x³ + ax (quartic twists); y² = x³ − n²x (congruent-number curves)
conductor (N): 5107679660790605376
discriminant (Δ): 2082049167740946703750131758091409814554127109358915584
Faltings height: 9.1122
naive height: 132.5277
primes of bad reduction: 2, 3, 19, 83, 211, 283
regulator: 133439.462873104010981247304043434475284161382774
submitted by: David Renshaw
submitted at: 2026-10-21 01:49:55 UTC
last updated: 2026-10-21 01:49:55 UTC

Witness: 5 independent points log in to add more points →

(-3492723, 1055906852481)
(6714746262, 548278856781264)
(-1466398060662/9409, 6187805727728388192/912673)
(26522265129/16, 4060676758380075/64)
(354725604844/625, 20030236155556172/15625)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 565004406. Found by a Mestre-Nagao sieve over n <= 3*10^9 (n = 5,6,7 mod 8), top candidates checked with ellrank.

last edited by David Renshaw at 2026-10-21 01:49:55 UTC · history