curve #6192

y² + xy + y = x³ − 27080046752958965321914029444754098x + 1716659733537306173794629481425989867022234632842756

a-invariants: [1, 0, 1, -27080046752958965321914029444754098, 1716659733537306173794629481425989867022234632842756]
c-invariants: [1299842244142030335451873413348196681, -1483194009776232536108323238165100748285020842798436421]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/3ℤ
conductor (N): 1088418986898974644212203666448533649270
discriminant (Δ): -2120483669548268809643076067393904874073985514018051267791303059465910159981149182843586888611375000000
Faltings height: 18.5558
naive height: 249.4676
primes of bad reduction: 2, 3, 5, 13, 17, 19, 29, 31, 43, 47, 223, 229, 5431, 17419, 1135429, 45629533
regulator: 53005270146.9199750663210902108056884347479359861089503860972
submitted by: Andrii Prokhorenkov
submitted at: 2026-10-30 15:40:31 UTC
last updated: 2026-10-30 15:40:31 UTC

Witness: 11 independent points log in to add more points →

(49591047396263045, 22264090455862809278866977)
(60756941362367645, 17194111000121477123512977)
(87613088443631717, 4076264920600935345829569)
(345107777068479170, 182956952458782711336911352)
(-1007659500004506347536/5329, 3398597639188406867601180863043/389017)
(-21870125232552924595/2809, 6535387967258597444288714969829/148877)
(2234109198068729477/16, 1641486554202794236752178887/64)
(16866764003067896600/121, 33944626528905167535739397337/1331)
(113424579416482279730/5929, 15851670445308045341079643598616/456533)
(427667385634837379620/4761, 982451870449485627362240311043/328509)
(581255345444249336165/529, 13864737657240813805157324868879/12167)

Commentary

Specialization at T = 2914/145 of the elliptic fibration of X648 given by: y^2 + (Lx+B)y = x^3, where L = 5098973881T^2 - 32101629380T + 51085836400, B = -2550648011199324171649113600T^5 - 231394683896910015292127210400T^4 + 1324630129545679262439545334000T^3 - 2330371426153230648960353010000T^2 + 1239686628515660725984584000000T.

last edited by Andrii Prokhorenkov at 2026-10-30 15:40:31 UTC · history