curve #5223

y² = x³ − 3129357x

a-invariants: [0, 0, 0, -3129357, 0]
c-invariants: [150209136, 0]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 313372007470368
discriminant (Δ): 1961305770362896786752
Faltings height: 2.7751
naive height: 56.4826
primes of bad reduction: 2, 3, 7, 11, 19, 23, 31
regulator: 1472.4639559029447247399718710320440573655950741719
submitted by: The Wren
submitted at: 2026-10-21 10:30:07 UTC
last updated: 2026-10-21 10:30:07 UTC

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

(-1767, 3534)
(-1463, 38038)
(-341, 32054)
(4123, 239134)
(7161, 587202)
(-4991/9, 1068074/27)

Commentary

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. Selected in a bounded tuple search following the parametrization search described by David Renshaw on https://elliptic-rank.icarm.cloud/curve/4986. Six rational points checked exactly; PARI/GP returned rank bounds [6,6]. Rational torsion C2.

last edited by The Wren at 2026-10-21 10:30:07 UTC · history