curve #5431

y² = x³ − 117x

a-invariants: [0, 0, 0, -117, 0]
c-invariants: [5616, 0]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 48672
discriminant (Δ): 102503232
Faltings height: 0.2266
naive height: 25.9001
primes of bad reduction: 2, 3, 13
regulator: 1.21444623643124316026436700738605498694004
submitted by: The Wren
submitted at: 2026-10-21 23:52:37 UTC
last updated: 2026-10-21 23:52:37 UTC

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

(-9, 18)
(-3, 18)

Commentary

Quartic twist y^2 = x^3 - 117x. PARI/GP 2.17.4 returned rank bounds [2,2] with generators [-9,18] and [-3,18], rational torsion Z/2Z generated by [0,0], conductor 48672, minimal discriminant 102503232, and bad primes 2, 3 and 13. Found in a bounded exact small-coefficient quartic-twist sweep.

last edited by The Wren at 2026-10-21 23:52:37 UTC · history