def __init__(self, parent, ideal, element=None):
"""
Returns the ideal class of this fractional ideal.
EXAMPLE::
sage: K.<a> = NumberField(x^2 + 23,'a'); G = K.class_group()
sage: G(K.ideal(13, a + 4))
Fractional ideal class (13, 1/2*a + 17/2)
"""
self.__ideal = ideal
if element is None:
element = map(int, ideal.ideal_class_log(proof=parent._proof_flag))
AbelianGroupElement.__init__(self, parent, element)
def __init__(self, parent, ideal, element=None):
"""
Returns the ideal class of this fractional ideal.
EXAMPLE::
sage: K.<a> = NumberField(x^2 + 23,'a'); G = K.class_group()
sage: G(K.ideal(13, a + 4))
Fractional ideal class (13, 1/2*a + 17/2)
"""
self.__ideal = ideal
if element is None:
element = map(int, ideal.ideal_class_log(proof=parent._proof_flag))
AbelianGroupElement.__init__(self, parent, element)
def __init__(self, exponents, parent, value=None):
"""
Create an element
EXAMPLES::
sage: F = AbelianGroupWithValues([1,-1], [2,4])
sage: a,b = F.gens()
sage: a*b^-1 in F
True
sage: (a*b^-1).value()
-1
"""
self._value = value
AbelianGroupElement.__init__(self, exponents, parent)
def __init__(self, parent, exponents, value=None):
"""
Create an element
EXAMPLES::
sage: F = AbelianGroupWithValues([1,-1], [2,4])
sage: a,b = F.gens()
sage: a*b^-1 in F
True
sage: (a*b^-1).value()
-1
"""
self._value = value
AbelianGroupElement.__init__(self, parent, exponents)
def _mul_(self, other):
r"""
Multiplication of two (S-)ideal classes.
EXAMPLES::
sage: G = NumberField(x^2 + 23,'a').class_group(); G
Class group of order 3 with structure C3 of Number Field in a with defining polynomial x^2 + 23
sage: I = G.0; I
Fractional ideal class (2, 1/2*a - 1/2)
sage: I*I # indirect doctest
Fractional ideal class (2, 1/2*a + 1/2)
sage: I*I*I # indirect doctest
Trivial principal fractional ideal class
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)*CS(G)
Trivial S-ideal class
"""
m = AbelianGroupElement._mul_(self, other)
m._value = (self.ideal() * other.ideal()).reduce_equiv()
return m
def _mul_(self, other):
r"""
Multiplication of two (S-)ideal classes.
EXAMPLE::
sage: G = NumberField(x^2 + 23,'a').class_group(); G
Class group of order 3 with structure C3 of Number Field in a with defining polynomial x^2 + 23
sage: I = G.0; I
Fractional ideal class (2, 1/2*a - 1/2)
sage: I*I # indirect doctest
Fractional ideal class (2, 1/2*a + 1/2)
sage: I*I*I # indirect doctest
Trivial principal fractional ideal class
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)*CS(G)
Trivial S-ideal class
"""
m = AbelianGroupElement._mul_(self, other)
m._value = (self.ideal() * other.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)
"""
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 __init__(self, parent, ideal, element=None):
r"""
Returns the S-ideal class of this fractional ideal.
EXAMPLES::
sage: K.<a> = QuadraticField(-14)
sage: I = K.ideal(2,a)
sage: S = (I,)
sage: CS = K.S_class_group(S)
sage: J = K.ideal(7,a)
sage: G = K.ideal(3,a+1)
sage: CS(I)
Trivial S-ideal class
sage: CS(J)
Trivial S-ideal class
sage: CS(G)
Fractional S-ideal class (3, a + 1)
"""
self.__ideal = ideal
if element is None:
element = ideal.S_ideal_class_log(parent.S())
AbelianGroupElement.__init__(self, parent, element)
def __init__(self, parent, ideal, element=None):
r"""
Returns the S-ideal class of this fractional ideal.
EXAMPLES::
sage: K.<a> = QuadraticField(-14)
sage: I = K.ideal(2,a)
sage: S = (I,)
sage: CS = K.S_class_group(S)
sage: J = K.ideal(7,a)
sage: G = K.ideal(3,a+1)
sage: CS(I)
Trivial S-ideal class
sage: CS(J)
Trivial S-ideal class
sage: CS(G)
Fractional S-ideal class (3, a + 1)
"""
self.__ideal = ideal
if element is None:
element = ideal.S_ideal_class_log(parent.S())
AbelianGroupElement.__init__(self, parent, element)
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"""
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 __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 _mul_(left, right):
"""
Multiply ``left`` and ``right``
TESTS::
sage: G.<a,b> = AbelianGroupWithValues([5,2], 2)
sage: a._mul_(b)
a*b
sage: a*b
a*b
sage: (a*b).value()
10
"""
m = AbelianGroupElement._mul_(left, right)
m._value = left.value() * right.value()
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 _mul_(self, other):
r"""
Multiplies together two S-ideal classes.
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)*CS(G)
Trivial S-ideal class
"""
m = AbelianGroupElement._mul_(self, other)
return SFractionalIdealClass(self.parent(), (self.ideal() * other.ideal()).reduce_equiv(), m.list())
def _mul_(self, other):
r"""
Multiplication of two ideal classes.
EXAMPLE::
sage: G = NumberField(x^2 + 23,'a').class_group(); G
Class group of order 3 with structure C3 of Number Field in a with defining polynomial x^2 + 23
sage: I = G.0; I
Fractional ideal class (2, 1/2*a - 1/2)
sage: I*I # indirect doctest
Fractional ideal class (2, 1/2*a + 1/2)
sage: I*I*I # indirect doctest
Trivial principal fractional ideal class
"""
m = AbelianGroupElement._mul_(self, other)
return FractionalIdealClass(self.parent(), (self.__ideal * other.__ideal).reduce_equiv(), m.list())
def _div_(left, right):
"""
Divide ``left`` by ``right``
TESTS::
sage: G.<a,b> = AbelianGroupWithValues([5,2], 2)
sage: a._div_(b)
a*b^-1
sage: a/b
a*b^-1
sage: (a/b).value()
5/2
"""
m = AbelianGroupElement._div_(left, right)
m._value = left.value() / right.value()
return m
def _mul_(left, right):
"""
Multiply ``left`` and ``right``
TESTS::
sage: G.<a,b> = AbelianGroupWithValues([5,2], 2)
sage: a._mul_(b)
a*b
sage: a*b
a*b
sage: (a*b).value()
10
"""
m = AbelianGroupElement._mul_(left, right)
m._value = left.value() * right.value()
return m
def _div_(left, right):
"""
Divide ``left`` by ``right``
TESTS::
sage: G.<a,b> = AbelianGroupWithValues([5,2], 2)
sage: a._div_(b)
a*b^-1
sage: a/b
a*b^-1
sage: (a/b).value()
5/2
"""
m = AbelianGroupElement._div_(left, right)
m._value = left.value() / right.value()
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 _div_(self, other):
r"""
Division of two ideal classes.
EXAMPLES::
sage: G = NumberField(x^2 + 23,'a').class_group(); G
Class group of order 3 with structure C3 of Number Field in a with defining polynomial x^2 + 23
sage: I = G.0; I
Fractional ideal class (2, 1/2*a - 1/2)
sage: I*I # indirect doctest
Fractional ideal class (2, 1/2*a + 1/2)
sage: I*I*I # indirect doctest
Trivial principal fractional ideal class
"""
m = AbelianGroupElement._div_(self, other)
m._value = (self.ideal() / other.ideal()).reduce_equiv()
return m
def _div_(self, other):
r"""
Division of two ideal classes.
EXAMPLE::
sage: G = NumberField(x^2 + 23,'a').class_group(); G
Class group of order 3 with structure C3 of Number Field in a with defining polynomial x^2 + 23
sage: I = G.0; I
Fractional ideal class (2, 1/2*a - 1/2)
sage: I*I # indirect doctest
Fractional ideal class (2, 1/2*a + 1/2)
sage: I*I*I # indirect doctest
Trivial principal fractional ideal class
"""
m = AbelianGroupElement._div_(self, other)
m._value = (self.ideal() / other.ideal()).reduce_equiv()
return m
def _mul_(self, other):
r"""
Multiplication of two ideal classes.
EXAMPLE::
sage: G = NumberField(x^2 + 23,'a').class_group(); G
Class group of order 3 with structure C3 of Number Field in a with defining polynomial x^2 + 23
sage: I = G.0; I
Fractional ideal class (2, 1/2*a - 1/2)
sage: I*I # indirect doctest
Fractional ideal class (2, 1/2*a + 1/2)
sage: I*I*I # indirect doctest
Trivial principal fractional ideal class
"""
m = AbelianGroupElement._mul_(self, other)
return FractionalIdealClass(
self.parent(), (self.__ideal * other.__ideal).reduce_equiv(),
m.list())
def _mul_(self, other):
r"""
Multiplies together two S-ideal classes.
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)*CS(G)
Trivial S-ideal class
"""
m = AbelianGroupElement._mul_(self, other)
return SFractionalIdealClass(
self.parent(), (self.ideal() * other.ideal()).reduce_equiv(),
m.list())
def order(self):
r"""
Return the order of this ideal class in the class group.
EXAMPLE::
sage: K.<w>=QuadraticField(-23)
sage: OK=K.ring_of_integers()
sage: C=OK.class_group()
sage: [c.order() for c in C]
[1, 3, 3]
sage: k.<a> = NumberField(x^2 + 20072); G = k.class_group(); G
Class group of order 76 with structure C38 x C2 of Number Field in a with defining polynomial x^2 + 20072
sage: [c.order() for c in G.gens()]
[38, 2]
"""
# an old method with a new docstring
return AbelianGroupElement.order(self)
def order(self):
r"""
Return the order of this ideal class in the class group.
EXAMPLE::
sage: K.<w>=QuadraticField(-23)
sage: OK=K.ring_of_integers()
sage: C=OK.class_group()
sage: [c.order() for c in C]
[1, 3, 3]
sage: k.<a> = NumberField(x^2 + 20072); G = k.class_group(); G
Class group of order 76 with structure C38 x C2 of Number Field in a with defining polynomial x^2 + 20072
sage: [c.order() for c in G.gens()]
[38, 2]
"""
# an old method with a new docstring
return AbelianGroupElement.order(self)
def __pow__(self, n):
"""
Exponentiate ``self``
INPUT:
- ``n`` -- integer. The exponent.
TESTS::
sage: G.<a,b> = AbelianGroupWithValues([5,2], 2)
sage: a^3
a^3
sage: (a^3).value()
125
"""
m = Integer(n)
if n != m:
raise TypeError('argument n (= '+str(n)+') must be an integer.')
pow_self = AbelianGroupElement.__pow__(self, m)
pow_self._value = pow(self.value(), m)
return pow_self
def __pow__(self, n):
"""
Exponentiate ``self``
INPUT:
- ``n`` -- integer. The exponent.
TESTS::
sage: G.<a,b> = AbelianGroupWithValues([5,2], 2)
sage: a^3
a^3
sage: (a^3).value()
125
"""
m = Integer(n)
if n != m:
raise TypeError('argument n (= '+str(n)+') must be an integer.')
pow_self = AbelianGroupElement.__pow__(self, m)
pow_self._value = pow(self.value(), m)
return pow_self
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
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))