curve #1166

y² = x³ + x² − 23677614453500806438013384095x + 1402345514268129069082402138467169596598250

a-invariants: [0, 1, 0, -23677614453500806438013384095, 1402345514268129069082402138467169596598250]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/4ℤ
conductor (N): 1128310354841987032068083640
discriminant (Δ): 3243939912551856986766443772872670662598800835538825751615730275341948376168400
Faltings height: 14.7375
naive height: 207.6166
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 23, 31, 41, 53, 61, 67, 109, 257, 263, 6857
regulator: 233557909.634031665003976922249867242089139925297950721
submitted by: David Renshaw
submitted at: 2026-09-23 21:38:07 UTC
last updated: 2026-09-23 21:38:07 UTC

Witness: 8 independent points log in to add more points →

(107419960239779, -313721489863986000615)
(216240168419152730/2209, -15598599911447018011100100/103823)
(32121862670251235/361, -15707124583829321430645/6859)
(114004798064435, -429784753869729330345)
(2041095956766964662515/22972849, -13648826374909369892756206335/110108865257)
(88849144792595, -140771353409894025)
(19107690969364505555/214369, -477627079987455322687410375/99252847)
(40709650599438560679467/458088409, 4547796063044293821810079028907/9804466217827)

Commentary

Obtained by computing a rational isogeny from curve#1108 and mapping all 8 independent witness points to the global minimal model. Isogenies preserve the Mordell-Weil rank; the image points are independently certified by the site verifier. Torsion computed exactly with PARI/GP elltors. Conductor is unchanged by the isogeny.

last edited by David Renshaw at 2026-09-23 21:38:07 UTC · history