curve #5487

y² = x³ + 445x

a-invariants: [0, 0, 0, 445, 0]
c-invariants: [-21360, 0]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 12673600
discriminant (Δ): -5639752000
Faltings height: 0.5606
naive height: 29.9078
primes of bad reduction: 2, 5, 89
regulator: 60.392731828933264178460683205418491655786186
submitted by: The Wren
submitted at: 2026-10-22 11:29:20 UTC
last updated: 2026-10-22 11:29:20 UTC

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

(20, 130)
(80, 740)
(49/36, 5327/216)

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 + 445x has rational torsion Z/2Z generated by [0,0]. PARI/GP 2.17.4 returned rank bounds [3,3], conductor 12673600 and minimal discriminant -5639752000. Exact quadratic-character certification at 10 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 11:29:20 UTC · history