def __call__(self, u):
"""
Returns the abstract group element corresponding to the unit u.
INPUT:
- ``u`` -- Any object from which an element of the unit group's number
field `K` may be constructed; an error is raised if an element of `K`
cannot be constructed from u, or if the element constructed is not a
unit.
EXAMPLES::
sage: x = polygen(QQ)
sage: K.<a> = NumberField(x^2-38)
sage: UK = UnitGroup(K)
sage: UK(1)
1
sage: UK(-1)
u0
sage: UK.gens()
[-1, 6*a - 37]
sage: UK.ngens()
2
sage: [UK(u) for u in UK.gens()]
[u0, u1]
sage: [UK(u).list() for u in UK.gens()]
[[1, 0], [0, 1]]
sage: UK(a)
Traceback (most recent call last):
...
ValueError: a is not a unit
"""
K = self.__number_field
try:
u = K(u)
except TypeError:
raise ValueError, "%s is not an element of %s"%(u,K)
if not u.is_integral() or u.norm().abs() != 1:
raise ValueError, "%s is not a unit"%u
m = K.pari_bnf().bnfisunit(pari(u)).mattranspose()
# convert column matrix to a list:
m = [ZZ(m[0,i].python()) for i in range(m.ncols())]
# NB pari puts the torsion at the end!
m.insert(0,m.pop())
return AbelianGroupElement(self, m)
def _element_constructor_(self, *args, **kwargs):
try:
return AbelianGroupElement(args[0], self)
except:
I = self.__number_field.ideal(*args, **kwargs)
return AbelianGroupElement(self, self.log(I))