def _element_constructor_(self, elt):
r"""
TESTS::
sage: UCF = UniversalCyclotomicField()
sage: UCF(3)
3
sage: UCF(3/2)
3/2
sage: C = CyclotomicField(13)
sage: UCF(C.gen())
E(13)
sage: UCF(C.gen() - 3*C.gen()**2 + 5*C.gen()**5)
E(13) - 3*E(13)^2 + 5*E(13)^5
sage: C = CyclotomicField(12)
sage: zeta12 = C.gen()
sage: a = UCF(zeta12 - 3* zeta12**2)
sage: a
-E(12)^7 + 3*E(12)^8
sage: C(_) == a
True
sage: UCF('[[0, 1], [0, 2]]')
Traceback (most recent call last):
...
TypeError: [ [ 0, 1 ], [ 0, 2 ] ]
of type <type 'sage.libs.gap.element.GapElement_List'> not valid
to initialize an element of the universal cyclotomic field
Some conversions from symbolic functions are possible::
sage: UCF = UniversalCyclotomicField()
sage: [UCF(sin(pi/k, hold=True)) for k in range(1,10)]
[0,
1,
-1/2*E(12)^7 + 1/2*E(12)^11,
1/2*E(8) - 1/2*E(8)^3,
-1/2*E(20)^13 + 1/2*E(20)^17,
1/2,
-1/2*E(28)^19 + 1/2*E(28)^23,
1/2*E(16)^3 - 1/2*E(16)^5,
-1/2*E(36)^25 + 1/2*E(36)^29]
sage: [UCF(cos(pi/k, hold=True)) for k in range(1,10)]
[-1,
0,
1/2,
1/2*E(8) - 1/2*E(8)^3,
-1/2*E(5)^2 - 1/2*E(5)^3,
-1/2*E(12)^7 + 1/2*E(12)^11,
-1/2*E(7)^3 - 1/2*E(7)^4,
1/2*E(16) - 1/2*E(16)^7,
-1/2*E(9)^4 - 1/2*E(9)^5]
.. TODO::
Implement conversion from QQbar (and as a consequence from the
symbolic ring)
"""
elt = py_scalar_to_element(elt)
if isinstance(elt, (Integer, Rational)):
return self.element_class(self, libgap(elt))
elif isinstance(
elt,
(GapElement_Integer, GapElement_Rational, GapElement_Cyclotomic)):
return self.element_class(self, elt)
elif not elt:
return self.zero()
obj = None
if isinstance(elt, gap.GapElement):
obj = libgap(elt)
elif isinstance(elt, gap3.GAP3Element):
obj = libgap.eval(str(elt))
elif isinstance(elt, str):
obj = libgap.eval(elt)
if obj is not None:
if not isinstance(obj, (GapElement_Integer, GapElement_Rational,
GapElement_Cyclotomic)):
raise TypeError(
"{} of type {} not valid to initialize an element of the universal cyclotomic field"
.format(obj, type(obj)))
return self.element_class(self, obj)
# late import to avoid slowing down the above conversions
from sage.rings.number_field.number_field_element import NumberFieldElement
from sage.rings.number_field.number_field import NumberField_cyclotomic, CyclotomicField
P = parent(elt)
if isinstance(elt, NumberFieldElement) and isinstance(
P, NumberField_cyclotomic):
n = P.gen().multiplicative_order()
elt = CyclotomicField(n)(elt)
return sum(c * self.gen(n, i)
for i, c in enumerate(elt._coefficients()))
if hasattr(elt, '_algebraic_'):
return elt._algebraic_(self)
raise TypeError(
"{} of type {} not valid to initialize an element of the universal cyclotomic field"
.format(elt, type(elt)))
