curve #3544

y² = x³ − x² − 54050310909963728401644728615x + 4819942348608245487926058303845433045684912

a-invariants: [0, -1, 0, -54050310909963728401644728615, 4819942348608245487926058303845433045684912]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 6431096057258181075868490059406440855350716841273480 ☆ record for ℤ/2ℤ torsion, rank ≥ 16
discriminant (Δ): 69733207919230225346859893902227312799325722773611271024930019201766848964603989218000 ☆ record for ℤ/2ℤ torsion, rank ≥ 16
Faltings height: 15.3286 ☆ record for ℤ/2ℤ torsion, rank ≥ 16
naive height: 210.0927 ☆ record for ℤ/2ℤ torsion, rank ≥ 16
primes of bad reduction: 2, 3, 5, 7, 17, 37, 59, 61, 73, 79, 89, 107, 131, 157, 179, 223, 1033, 1193, 12203, 22433, 47581, 4672963
regulator: 4473014018819502.685015379771718422326321943835741636951753147834881527
submitted by: wgxli
submitted at: 2026-09-28 01:05:09 UTC
last updated: 2026-09-28 01:05:09 UTC

Witness: 16 independent points log in to add more points →

(126613958612677, 78605709477053359031)
(126703101785369, 75172124935048954635)
(127367562748612, 43655063643097152446)
(141015680201497, 46446838444668572939)
(89333423370509, 839264499485542116945)
(-268323272731651, 65775269567203584255)
(101620337991364, 613786223282860999490)
(163481592273269, 594102903140805699015)
(169316853158624, 722718932104725787980)
(171276106322817, 766085213810566234941)
(20292213025009, 1931708500122117493805)
(185429009645284, 1083153967444350230330)
(200312515207769, 1424973360234786706485)
(366662066402957, 5856294437655705405999)
(-125153095971811, 3102288821269058197665)
(2763673135308193, 144789750118517665822045)

Commentary

Construction adapted from S. Kihara, Proc. Japan Acad., 77, Ser. A (2001). Parameters (a, b, c, u) = (408, 1129, 1166, 556). With apologies to @mathandcobb!

last edited by wgxli at 2026-09-28 01:05:09 UTC · history