def inverse(self):
r"""
Return the multiplicative inverse of this ideal class.
EXAMPLE::
sage: K.<a> = NumberField(x^3 - 3*x + 8); G = K.class_group()
sage: G(2, a).inverse()
Fractional ideal class (2, a^2 + 2*a - 1)
"""
m = AbelianGroupElement.inverse(self)
return FractionalIdealClass(self.parent(), (~self.__ideal).reduce_equiv(), m.list())
def inverse(self):
r"""
Return the multiplicative inverse of this ideal class.
EXAMPLE::
sage: K.<a> = NumberField(x^3 - 3*x + 8); G = K.class_group()
sage: G(2, a).inverse()
Fractional ideal class (2, a^2 + 2*a - 1)
"""
m = AbelianGroupElement.inverse(self)
return FractionalIdealClass(self.parent(),
(~self.__ideal).reduce_equiv(), m.list())
def inverse(self):
r"""
Return the multiplicative inverse of this ideal class.
EXAMPLES::
sage: K.<a> = NumberField(x^3 - 3*x + 8); G = K.class_group()
sage: G(2, a).inverse()
Fractional ideal class (2, a^2 + 2*a - 1)
sage: ~G(2, a)
Fractional ideal class (2, a^2 + 2*a - 1)
"""
m = AbelianGroupElement.inverse(self)
m._value = (~self.ideal()).reduce_equiv()
return m
def inverse(self):
r"""
Return the multiplicative inverse of this ideal class.
EXAMPLE::
sage: K.<a> = NumberField(x^3 - 3*x + 8); G = K.class_group()
sage: G(2, a).inverse()
Fractional ideal class (2, a^2 + 2*a - 1)
sage: ~G(2, a)
Fractional ideal class (2, a^2 + 2*a - 1)
"""
m = AbelianGroupElement.inverse(self)
m._value = (~self.ideal()).reduce_equiv()
return m
def inverse(self):
r"""
Finds the inverse of the given S-ideal class.
EXAMPLES::
sage: K.<a> = QuadraticField(-14)
sage: I = K.ideal(2,a)
sage: S = (I,)
sage: CS = K.S_class_group(S)
sage: G = K.ideal(3,a+1)
sage: CS(G).inverse()
Fractional S-ideal class (3, a + 2)
"""
m = AbelianGroupElement.inverse(self)
inv_ideal = self.parent().number_field().class_group()(self.ideal()).inverse().ideal()
return SFractionalIdealClass(self.parent(), inv_ideal , m.list())
def inverse(self):
r"""
Finds the inverse of the given S-ideal class.
EXAMPLES::
sage: K.<a> = QuadraticField(-14)
sage: I = K.ideal(2,a)
sage: S = (I,)
sage: CS = K.S_class_group(S)
sage: G = K.ideal(3,a+1)
sage: CS(G).inverse()
Fractional S-ideal class (3, a + 2)
"""
m = AbelianGroupElement.inverse(self)
inv_ideal = self.parent().number_field().class_group()(
self.ideal()).inverse().ideal()
return SFractionalIdealClass(self.parent(), inv_ideal, m.list())
def inverse(self):
"""
Return the inverse element.
EXAMPLE::
sage: G.<a,b> = AbelianGroupWithValues([2,-1], [0,4])
sage: a.inverse()
a^-1
sage: a.inverse().value()
1/2
sage: a.__invert__().value()
1/2
sage: (~a).value()
1/2
sage: (a*b).value()
-2
sage: (a*b).inverse().value()
-1/2
"""
m = AbelianGroupElement.inverse(self)
m._value = ~self.value()
return m
def inverse(self):
"""
Return the inverse element.
EXAMPLE::
sage: G.<a,b> = AbelianGroupWithValues([2,-1], [0,4])
sage: a.inverse()
a^-1
sage: a.inverse().value()
1/2
sage: a.__invert__().value()
1/2
sage: (~a).value()
1/2
sage: (a*b).value()
-2
sage: (a*b).inverse().value()
-1/2
"""
m = AbelianGroupElement.inverse(self)
m._value = ~self.value()
return m
