curve #3523

y² + xy + y = x³ + x² − 92167809678670358286515216060475x + 338229404052485673037798099794982148230278297801

a-invariants: [1, 1, 1, -92167809678670358286515216060475, 338229404052485673037798099794982148230278297801]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 1789399932124511774089607156478992325365460369832460586
discriminant (Δ): 688811158542019060834308601091705462064656657898351677864385828264770040417445265052875828232192
Faltings height: 17.2174
naive height: 232.4171
primes of bad reduction: 2, 3, 7, 11, 23, 43, 53, 71, 101, 107, 113, 193, 283, 491, 509, 1361, 2287, 4057, 6899, 23857, 49157, 611071
regulator: 216578685366554.5002217442841867788528887379853764341766208979415800566
submitted by: Alvaro Lozano-Robledo
submitted at: 2026-09-27 19:17:11 UTC
last updated: 2026-09-27 19:17:11 UTC

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

(443143539999749670363/85849, -72745942040543879618437410962/25153757)
(7683033234179123, 289176111134034933330342)
(111427202957698098067/28561, -941288747494181466140939572714/4826809)
(211134308963918142278192957/60620348944, -3639733087892304426696556003166849760543/14925457354200128)
(-686424451074385512583906717/160726422649, -52108254452324127920658722732348593910358/64436347924942643)
(-72039890365665648053162529781738565/6575750750747283561, 3067488307779126029071343346528048105480141473971438/16862346715835027925643344309)
(3282468953710899, -266568432297571746267226)
(7841507243963059, 312511028198727514754726)
(2370469800236743, -364785370529246516195218)
(8212828764512863, 367737449315595962025122)
(8106438902390803, 351832431936800009900294)
(5016187740162727, -46004159958795810841942)
(8434241125764659, 401055628700939056792614)
(6486068728963567, 115263387029161214540594)
(-362820708133781921111261/197993041, -1971886401813430315798162075338262966/2785960079911)
(2100041037976153988418011467/90113435721, -89091447691632383533639508887120783562498/27051062155651269)

Commentary

Rank exactly 16, torsion Z/2Z. Specialization at T = −1247/882 of the rank-9 fibration of X400 with a 2-torsion point (Li, 2026), and its 2-isogeny pair.

last edited by Alvaro Lozano-Robledo at 2026-09-27 19:17:11 UTC · history