def base_extend(self, S):
r"""
Returns the conic over ``S`` given by the same equation as ``self``.
EXAMPLES::
sage: c = Conic([1, 1, 1]); c
Projective Conic Curve over Rational Field defined by x^2 + y^2 + z^2
sage: c.has_rational_point()
False
sage: d = c.base_extend(QuadraticField(-1, 'i')); d
Projective Conic Curve over Number Field in i with defining polynomial x^2 + 1 defined by x^2 + y^2 + z^2
sage: d.rational_point(algorithm = 'rnfisnorm')
(i : 1 : 0)
"""
if S in _Fields:
B = self.base_ring()
if B == S:
return self
if not S.has_coerce_map_from(B):
raise ValueError, "No natural map from the base ring of self " \
"(= %s) to S (= %s)" % (self, S)
from constructor import Conic
con = Conic([S(c) for c in self.coefficients()], \
self.variable_names())
if self._rational_point != None:
pt = [S(c) for c in Sequence(self._rational_point)]
if not pt == [0,0,0]:
# The following line stores the point in the cache
# if (and only if) there is no point in the cache.
pt = con.point(pt)
return con
return ProjectiveCurve_generic.base_extend(self, S)
def base_extend(self, S):
r"""
Returns the conic over ``S`` given by the same equation as ``self``.
EXAMPLES::
sage: c = Conic([1, 1, 1]); c
Projective Conic Curve over Rational Field defined by x^2 + y^2 + z^2
sage: c.has_rational_point()
False
sage: d = c.base_extend(QuadraticField(-1, 'i')); d
Projective Conic Curve over Number Field in i with defining polynomial x^2 + 1 defined by x^2 + y^2 + z^2
sage: d.rational_point(algorithm = 'rnfisnorm')
(i : 1 : 0)
"""
if S in _Fields:
B = self.base_ring()
if B == S:
return self
if not S.has_coerce_map_from(B):
raise ValueError("No natural map from the base ring of self " \
"(= %s) to S (= %s)" % (self, S))
from constructor import Conic
con = Conic([S(c) for c in self.coefficients()], \
self.variable_names())
if self._rational_point is not None:
pt = [S(c) for c in Sequence(self._rational_point)]
if not pt == [0,0,0]:
# The following line stores the point in the cache
# if (and only if) there is no point in the cache.
pt = con.point(pt)
return con
return ProjectiveCurve_generic.base_extend(self, S)
