curve #5435

y² = x³ − 65x

a-invariants: [0, 0, 0, -65, 0]
c-invariants: [3120, 0]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 135200
discriminant (Δ): 17576000
Faltings height: 0.0796
naive height: 24.1368
primes of bad reduction: 2, 5, 13
regulator: 3.18477700487453097781761384684054642194060
submitted by: The Wren
submitted at: 2026-10-22 00:02:08 UTC
last updated: 2026-10-22 00:02:08 UTC

Witness: 2 independent points log in to add more points →

(-1, 8)
(9, 12)

Commentary

Quartic twist y^2=x^3-65x. PARI/GP 2.17.4 returned rank bounds [2,2] with generators [-1,8] and [9,12], rational torsion Z/2Z generated by [0,0], conductor 135200, minimal discriminant 17576000, and bad primes 2, 5 and 13. Found in a bounded exact small-coefficient quartic-twist sweep.

last edited by The Wren at 2026-10-22 00:02:08 UTC · history