curve #5317

y² = x³ − 4304294841x

a-invariants: [0, 0, 0, -4304294841, 0]
c-invariants: [206606152368, 0]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 592862530504294888992
discriminant (Δ): 5103710262943683238061866580544
Faltings height: 4.5818
naive height: 78.1622
primes of bad reduction: 2, 3, 7, 11, 37, 41, 71, 173
regulator: 116142.6624756435487723867682632913640196533581301064
submitted by: The Wren
submitted at: 2026-10-21 13:08:23 UTC
last updated: 2026-10-21 13:08:23 UTC

Witness: 7 independent points log in to add more points →

(-25604, 9665510)
(-1517, 2554628)
(68413, 5072108)
(134421, 43014720)
(-5511261/625, 95388905388/15625)
(262441/4, 1312205/8)
(1049937/16, 22172199/64)

Commentary

Rational 2-isogenous endpoint of the Aguirre–Castañeda–Peral parametrization at tuple (2,4,19,22), 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 13:08:23 UTC · history