curve #5436

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): 270400
discriminant (Δ): -17576000
Faltings height: 0.0796
naive height: 24.1368
primes of bad reduction: 2, 5, 13
regulator: 10.6312842340253314507900915727765398023094
submitted by: The Wren
submitted at: 2026-10-22 00:02:46 UTC
last updated: 2026-10-22 00:02:46 UTC

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

(4, 18)
(1040, 33540)

Commentary

Quartic twist y^2=x^3+65x. PARI/GP 2.17.4 returned rank bounds [2,2] with generators [4,18] and [1040,33540], rational torsion Z/2Z generated by [0,0], conductor 270400, 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 00:02:46 UTC · history