curve #5022

y² = x³ − 6786025x

a-invariants: [0, 0, 0, -6786025, 0]
c-invariants: [325729200, 0]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
families: y² = x³ + ax (quartic twists); y² = x³ − n²x (congruent-number curves)
conductor (N): 217152800
discriminant (Δ): 19999831641819121000000
Faltings height: 2.9686
naive height: 58.8047
primes of bad reduction: 2, 5, 521
regulator: 412.03206627638600168439935904451693551513159
submitted by: David Renshaw
submitted at: 2026-10-21 01:48:01 UTC
last updated: 2026-10-21 01:48:01 UTC

Witness: 3 independent points log in to add more points →

(-1805, 79800)
(99405, 31330200)
(-18605/9, 1945900/27)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 2605. Found in an exhaustive ellrank scan of squarefree n <= 20000; one of the smallest n of rank 3.

last edited by David Renshaw at 2026-10-21 01:48:01 UTC · history