curve #755

y² = x³ + x² − 6600328628343578828177865x + 6638285365080080725945250624876486400

a-invariants: [0, 1, 0, -6600328628343578828177865, 6638285365080080725945250624876486400]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 26659116102684463027381808015526996103613420424556860
discriminant (Δ): -634379051551098740963024306612601755354974329742296970970539374600780750000
Faltings height: 13.1459
naive height: 183.0950
primes of bad reduction: 2, 3, 5, 89, 131, 487, 446969, 1302233, 40737673453, 3300212536077131170697
regulator: 790998792615258.678786386836010201264547967106504102035749075850217746225
submitted by: Valery Asiryan
submitted at: 2026-09-18 01:24:04 UTC
last updated: 2026-09-18 01:24:04 UTC

Witness: 17 independent points log in to add more points →

(-2831987151655, -1617827558075970075)
(-2584906219227, -2535319345744942899)
(-2206761559360, -3233752768363594920)
(-1930569864441, -3490738312411986435)
(-1548569888067, -3625710765448651491)
(-991683897360, -3494061589818594420)
(-652001555635, -3265660745781476355)
(-263386787316, -2891098907259668010)
(388309328625, -2033191351746150795)
(506060990445, -1851409622707508925)
(658295042775, -1605800566981772355)
(41086411377261/25, -59925075730554082809/125)
(1201587635250345/16, -41627689693903321258275/64)
(3984075292620, -6601570443213838650)
(2154295109415, -1554762418806936195)
(22894630360497285/3721, -3201605278416107003328015/226981)
(858250243397745/256, -19295323179016885632375/4096)

Commentary

Specialization at T = -131/356 of the rank-11 fibration of X163 described by wgxli for ICARM #718. Independently recovered polynomial sections and rational-point search yield 17 points with an exact independence certificate, proving rank >= 17.

last edited by Valery Asiryan at 2026-09-18 01:24:05 UTC · history