curve #5486

y² = x³ + 579x

a-invariants: [0, 0, 0, 579, 0]
c-invariants: [-27792, 0]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 10727712
discriminant (Δ): -12422690496
Faltings height: 0.6264
naive height: 30.6975
primes of bad reduction: 2, 3, 193
regulator: 30.924784200162360768171168534411479720567943
submitted by: The Wren
submitted at: 2026-10-22 11:03:13 UTC
last updated: 2026-10-22 11:03:13 UTC

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

(3, 42)
(48, 372)
(147, 1806)

Commentary

Found in an exact bounded sweep of quartic twists y^2 = x^3 + Ax over every integer A with 301 <= |A| <= 1500. The model y^2 = x^3 + 579x has rational torsion Z/2Z generated by [0,0]. PARI/GP 2.17.4 returned rank bounds [3,3], conductor 10727712 and minimal discriminant -12422690496. Exact quadratic-character certification at 9 good primes gives an F_2 matrix of rank 4 with torsion-row rank 1, proving the three submitted points independent modulo torsion.

last edited by The Wren at 2026-10-22 11:03:13 UTC · history