curve #753

y² + xy + y = x³ − x² − 728462675385378432x + 239465662134141397669732515

a-invariants: [1, -1, 1, -728462675385378432, 239465662134141397669732515]
rank (lower bound): ≥ 14
torsion subgroup: trivial
conductor (N): 512298767178892797693771663917176742
discriminant (Δ): -32398123924122419285635681809128533642016068097036288
Faltings height: 9.0085
naive height: 135.0041
primes of bad reduction: 2, 7, 11, 13, 197, 19232441, 179326747, 376628804950709
regulator: 11095273810.2811108809858117480720166574922396427551539113820914934
submitted by: Valery Asiryan
submitted at: 2026-09-18 01:07:31 UTC
last updated: 2026-09-18 01:07:31 UTC

Witness: 14 independent points log in to add more points →

(-972729743, -5259078510155)
(-953616803, -8180909750175)
(-661401187, -20782854297983)
(-24303187, -16036049465983)
(409391525, -3139283805391)
(421152285, -2715243809055)
(434532421, -2230181245839)
(446386109, -1799441931295)
(449957765, -1669826525815)
(452246117, -1586915718935)
(476152461, -748708209103)
(179662461969/25, -75639037621064763/125)
(93452936789/121, -15610757338877485/1331)
(1978646685, -80899578120831)

Commentary

Specialization at T = 98/99 of the rank-11 fibration of X163 described by wgxli for ICARM #718. Independently recovered polynomial sections and rational-point search yield 14 points with an exact independence certificate, proving rank >= 14.

last edited by Valery Asiryan at 2026-09-18 01:07:31 UTC · history