curve #5338

y² = x³ − 63693123929206317x

a-invariants: [0, 0, 0, -63693123929206317, 0]
c-invariants: [3057269948601903216, 0]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 14424227683062166561921285520727072
discriminant (Δ): 16537034185206607303606415304597934961029543235648832
Faltings height: 8.7093
naive height: 127.6922
primes of bad reduction: 2, 3, 7, 13, 19, 23, 37, 73, 181, 313, 1163
regulator: 103057208.65136840848798296131100595995701175110302232112
submitted by: The Wren
submitted at: 2026-10-21 14:00:27 UTC
last updated: 2026-10-21 14:00:27 UTC

Witness: 9 independent points log in to add more points →

(-245250837, 932443682274)
(-83880212, 2180004769774)
(337319787, 4110578924382)
(457908633, 8176111278426)
(551858619, 11528998378794)
(560225241, 11838306309318)
(-492489450752/4761, 769139335935573616/328509)
(-18681091061/81, 1134577384498774/729)
(23825361049/36, 3399664593442859/216)

Commentary

Rational 2-isogeny endpoint from the Aguirre–Castañeda–Peral parametrization at tuple (3,5,20,27), 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.

last edited by The Wren at 2026-10-21 14:00:27 UTC · history