curve #5224

y² = x³ + 12517428x

a-invariants: [0, 0, 0, 12517428, 0]
c-invariants: [-600836544, 0]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 313372007470368
discriminant (Δ): -125523569303225394352128
Faltings height: 3.1217
naive height: 60.6415
primes of bad reduction: 2, 3, 7, 11, 19, 23, 31
regulator: 1472.4639559029447247399718710320440573655950741719
submitted by: The Wren
submitted at: 2026-10-21 10:31:23 UTC
last updated: 2026-10-21 10:31:23 UTC

Witness: 6 independent points log in to add more points →

(462, 76692)
(966, 113988)
(1254, 132924)
(2622, 225492)
(8694, 875196)
(27094/25, 15226828/125)

Commentary

Rational 2-isogeny partner of curve #5223: https://elliptic-rank.icarm.cloud/curve/5223. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (3,12,18,21), 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. Six rational points checked exactly; PARI/GP returned rank bounds [6,6]. Rational torsion C2.

last edited by The Wren at 2026-10-21 10:31:23 UTC · history