curve #6195

y² + xy = x³ − 645934870548626184826495245x + 6395226854140847719926321595597087851025

a-invariants: [1, 0, 0, -645934870548626184826495245, 6395226854140847719926321595597087851025]
c-invariants: [31004873786334056871671771761, -5525476001977738937327021359681191410943241]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/3ℤ
conductor (N): 186145258953442579858523828625110
discriminant (Δ): -420041500701545602610929772166893436418613189173303320298243398803968000000000000
Faltings height: 14.2758
naive height: 196.8359
primes of bad reduction: 2, 5, 11, 13, 17, 23, 31, 83, 131, 181, 3541, 1541074382026489
regulator: 97847839.29881679119505048772770377418699785496549948854588
submitted by: Andrii Prokhorenkov
submitted at: 2026-10-30 15:40:51 UTC
last updated: 2026-10-30 15:40:51 UTC

Witness: 10 independent points log in to add more points →

(-13153921001470, 112320252052687795455)
(-3843133357598, 93919535966063069887)
(90882204450, 79602283965902785855)
(6255848630640, 50982177403230897305)
(15052042729090, 9101555283667945855)
(16453381797890, 14885248598673741055)
(18652786616930, 28922980647864007615)
(34659550258370, 160135527184398053055)
(179168364603980, 2375331901991694300565)
(1384800148816610/49, 35500862383820757341865/343)

Commentary

Specialization at T = 5239/3439 of the elliptic fibration of X648 given by: y^2 + (Lx+B)y = x^3, where L = 3885067921T^2 - 14778309212T + 14171920816, B = 4936839685420991948803368000T^5 - 37762139727356747035718880000T^4 + 97681596151298882036363082000T^3 - 104946797327065126983361746000T^2 + 40090501217702000033914176000T.

last edited by Andrii Prokhorenkov at 2026-10-30 15:40:51 UTC · history