curve #5092

y² = x³ + 54468x

a-invariants: [0, 0, 0, 54468, 0]
c-invariants: [-2614464, 0]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 659280672
discriminant (Δ): -10341993497038848
Faltings height: 1.7624
naive height: 44.3297
primes of bad reduction: 2, 3, 17, 89
regulator: 38.742787245946291391085873824392325366818666
submitted by: The Wren
submitted at: 2026-10-21 02:36:10 UTC
last updated: 2026-10-21 02:36:10 UTC

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

(64, 1936)
(178, 3916)
(178/9, 28124/27)

Commentary

Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (3,9,33,45), reduced by rational fourth-power scaling. Reference: "High rank elliptic curves with torsion group Z/(2Z)", Math. Comp. 73 (2004), 323–331, doi:10.1090/S0025-5718-03-01547-3. Selected by height screening in a bounded tuple search, following the parametrization search described by David Renshaw on https://elliptic-rank.icarm.cloud/curve/4986. Three rational points checked exactly; PARI/GP returned rank bounds [3,3]. Rational torsion C2.

last edited by The Wren at 2026-10-21 02:36:10 UTC · history