curve #5098

y² = x³ + 6281780x

a-invariants: [0, 0, 0, 6281780, 0]
c-invariants: [-301525440, 0]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 78921519936800
discriminant (Δ): -15864564016274928128000
Faltings height: 2.9493
naive height: 58.5731
primes of bad reduction: 2, 5, 19, 61, 271
regulator: 870.8152118298025448803806764495857264404830562
submitted by: The Wren
submitted at: 2026-10-21 02:44:09 UTC
last updated: 2026-10-21 02:44:09 UTC

Witness: 4 independent points log in to add more points →

(61, 19581)
(196, 35196)
(2000, 143400)
(3904, 289872)

Commentary

Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (3,23,24,44), 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. Four rational points checked exactly; PARI/GP returned rank bounds [4,4]. Rational torsion C2.

last edited by The Wren at 2026-10-21 02:44:09 UTC · history