curve #5242

y² = x³ − 5591534025x

a-invariants: [0, 0, 0, -5591534025, 0]
c-invariants: [268393633200, 0]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 330739863830560800
discriminant (Δ): 11188526372296307665588881000000
Faltings height: 4.6472
naive height: 78.9472
primes of bad reduction: 2, 3, 5, 7, 11, 23, 43, 89
regulator: 42428.237200369877293585104188197041894054902082278
submitted by: The Wren
submitted at: 2026-10-21 10:56:22 UTC
last updated: 2026-10-21 10:56:22 UTC

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

(-1155, 2541000)
(-644, 1897546)
(307395, 165310200)
(641355, 510123900)
(4784395, 10463753300)
(-18975/4, 41118825/8)

Commentary

Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (5,6,12,20), 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 10:56:22 UTC · history