curve #5606

y² = x³ − 5372212109782332x + 158265389883843829423744

a-invariants: [0, 0, 0, -5372212109782332, 158265389883843829423744]
c-invariants: [257866181269551936, -136741296859641068622114816]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/4ℤ
conductor (N): 5585900418520662240
discriminant (Δ): -897788737734458824321263883512358475342400000000
Faltings height: 7.9609
naive height: 120.3603
primes of bad reduction: 2, 3, 5, 7, 11, 13, 23, 37, 107, 2213, 19231
regulator: 9056.726461884592206425345404632975817351509876880915
submitted by: The Wren
submitted at: 2026-10-23 22:27:30 UTC
last updated: 2026-10-23 22:27:30 UTC

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

(46282058, 93626186940)
(57333098, 196770279420)
(61777538, 249309274500)
(66319658, 306064822140)
(104063210, 852133814532)
(234396413, 3431788234875)
(308114288, 5268174219000)

Commentary

Obtained in my own run by specializing the Khoshnam–Moody four-section family with rational torsion Z/4Z at (c,d,m,n) = (3/11,1/11,1,1). Source: Sections 2–3, equations (2.1)–(3.3): https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=922952 The four specialized sections seeded the point search. Seven independent rational points were obtained and transported to the global minimal model [0,0,0,-5372212109782332,158265389883843829423744]. All seven points were checked exactly on this model. The computation returned matching rank bounds [7,7], establishing rank exactly 7, with rational torsion Z/4Z and conductor 5585900418520662240.

last edited by The Wren at 2026-10-23 22:27:30 UTC · history