Potential inconsistent results with fundamental units and KASH 3

[Georgi Guninski <guninski@guninski.com>]

Probably I am missing something, but get strange results.

In 2.5.1 via sage:

pr.<x>=ZZ[];K.<a>=NumberField(x^3 - 6*x^2 + 9*x + 1);K.unit_group().gens()
[-1, a^2 - 3*a]

in KASH 3:

f:=X^3 - 6*X^2 + 9*X + 1;nf:=NumberField(f);nu:=UnitGroup(nf);gn:=Generators(nu);  Apply(x->nu.ext1(x),List(Generators(nu))); 

[ [3, -1, 0], [-1, 0, 0] ] # 3 - a, -1

$-1$ is torsion in both cases.

Is this result consistent?

If g1= a^2 - 3*a and g2=3 - a I naiively expect to be able to solve either

(+/- g1)^n = +/- g2
or
(+/- g2)^n = +/- g1

Can't solve either for n<=10^4 and the coefficient are growing quite fast.

I suspect if a solution doesn't exist then rank will be greater than 1.

What am I missing?

Thanks.