curve #1136

y² = x³ + x² + 118581202588567310563728130640x + 3980688541421865908832460581640572647559008

a-invariants: [0, 1, 0, 118581202588567310563728130640, 3980688541421865908832460581640572647559008]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
conductor (N): 1128310354841987032068083640
discriminant (Δ): -113560926037211684803407333113505047703110088238583058026856495778931997572812500000000000
Faltings height: 15.7773
naive height: 212.4498
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 23, 31, 41, 53, 61, 67, 109, 257, 263, 6857
regulator: 59790824866.3121062410180920959660139748198208762753845
submitted by: David Renshaw
submitted at: 2026-09-23 21:37:11 UTC
last updated: 2026-09-23 21:37:11 UTC

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

(5214659183189899/9, 441349788112072855468750/27)
(91631075910969091/81, 29030385893943634923511250/729)
(905996088091705279/1089, -936064675494166647788897500/35937)
(181695539961568031611701181/401213295396, 3118285747119114456193015291293289398125/254134118289961944)
(358208889629899/9, -79928149832769347821250/27)
(458511607229899/9, -86036943515434214421250/27)
(107350398144202730260261/55860676, -35742600898558708737841355227370625/417502692424)
(104311761791152531/225, -42503582484330078229795754/3375)

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 21:37:11 UTC · history