curve #4774

y² = x³ − 93922872848724146729053666257x

a-invariants: [0, 0, 0, -93922872848724146729053666257, 0]
c-invariants: [4508297896738759042994575980336, 0]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 31365354823671479158815749670876719302406866939637832497952
discriminant (Δ): 53026636193258632499856653805489541121615449979092195130946204976111534453285788792101952
Faltings height: 15.7141
naive height: 211.7504
primes of bad reduction: 2, 3, 11, 13, 29, 31, 43, 47, 59, 79, 83, 173, 313, 373, 1487, 3457
regulator: 153433767881622.9688155881698244054234767371511103269461975890933
submitted by: David Renshaw
submitted at: 2026-10-20 21:29:57 UTC
last updated: 2026-10-20 21:29:57 UTC

Witness: 13 independent points log in to add more points →

(-276575640135201, -2195535978875023935084)
(-5723054432289, -733033603959375795948)
(-5507920096053921/25, -395259427531107434812608/125)
(-304653891626084, -581226593684433074378)
(23349585533550588/49, 2732165813058865821880866/343)
(1001302871976927, 30164026061532099012588)
(-844205963346119/16, -140344761205244805896467/64)
(-215810114867127, 3196617659767052691984)
(3935217139484887/9, 176074852891011071539868/27)
(7399454389697142972/6889, 19291238531557874575134506658/571787)
(-12110862831561, 1065697280305629027264)
(-2756408013641711/9, -5241628634863110963616/27)
(1817241281560031004639/3094081, -66084895405957994264159002407996/5442488479)

Commentary

Quartic twist y^2 = x^3 + a x with a = -93922872848724146729053666257. Source: Aguirre, Castaneda, Peral, 'High rank elliptic curves with torsion group Z/(2Z)', Math. Comp. 73 (2004) 323-331, doi:10.1090/S0025-5718-03-01547-3, https://www.ams.org/mcom/2004-73-245/S0025-5718-03-01547-3/.

last edited by David Renshaw at 2026-10-20 21:29:57 UTC · history