curve #3589

y² = x³ + x² − 43013639107094111303529305x + 108780634152014517781579619207784767799

a-invariants: [0, 1, 0, -43013639107094111303529305, 108780634152014517781579619207784767799]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 17346348574378923720434937981014495076047771460206208
discriminant (Δ): -18662262217531282821665525866223065740254630099610002247525389188656473776128
Faltings height: 13.5199
naive height: 188.6879
primes of bad reduction: 2, 3, 7, 11, 13, 17, 23, 31, 30325439, 354167135908081, 346648208624175386833
regulator: 476951495200295.04552246507726719265450428388009060658645382307704864999048
submitted by: wgxli
submitted at: 2026-09-28 22:51:51 UTC
last updated: 2026-09-28 22:51:51 UTC

Witness: 18 independent points log in to add more points →

(3543727205947, 924211106332426836)
(3544030174459, 923336019387273492)
(3680232644863, 570893722736637300)
(3797695607935, 447392549964391668)
(3812247507346, 454169082913050561)
(3819094980475, 459150423176577132)
(3822244803151, 461814428854142364)
(3850690555759, 495739921607602308)
(3877462711897, 541670823694064064)
(3958361763205, 734293654622339868)
(517670996617, 9308729660323397304)
(635837552971, 9038141927221997436)
(2302115818777, 4686028604589508176)
(4265107453381, 1705876600341477564)
(4454625714313, 2359488121283933640)
(5256539373145, 5284152706299584832)
(5766717429819, 7246067377232151468)
(13583890493563, 45066792785156312940)

Commentary

Specialization at T = 8 of the rank-17 fibration of X1008 given by: y^2 - (Lx+B)y = x(x+D)(x+E), where B = pq(L+p+q) - pE - qD, L = 94992T^2 + 43200T - 74684, p = 45650T^2 - 3480T + 2996, q = 7132T^2 - 71704T - 112100, D = 5(101326060T^4 + 455923735T^3 - 335745592T^2 - 107751313T - 99088206), E = 2(537101785T^4 - 5857182939T^3 - 5207884583T^2 + 312473955T - 77661626).

last edited by wgxli at 2026-09-29 00:33:36 UTC · history