def __init__(self, generator_orders, names, values, values_group):
"""
The Python constructor
TESTS::
sage: G = AbelianGroupWithValues([2,-1], [0,4]); G
Multiplicative Abelian group isomorphic to Z x C4
sage: cm = sage.structure.element.get_coercion_model()
sage: cm.explain(G, ZZ, operator.add)
Coercion on left operand via
Generic morphism:
From: Multiplicative Abelian group isomorphic to Z x C4
To: Integer Ring
Arithmetic performed after coercions.
Result lives in Integer Ring
Integer Ring
"""
self._values = values
self._values_group = values_group
AbelianGroup_class.__init__(self, generator_orders, names)
self._populate_coercion_lists_(embedding=self.values_embedding())
if self.ngens() != len(self._values):
raise ValueError('need one value per generator')
def __init__(self, generator_orders, names, values, values_group):
"""
The Python constructor
TESTS::
sage: G = AbelianGroupWithValues([2,-1], [0,4]); G
Multiplicative Abelian group isomorphic to Z x C4
sage: cm = sage.structure.element.get_coercion_model()
sage: cm.explain(G, ZZ, operator.add)
Coercion on left operand via
Generic morphism:
From: Multiplicative Abelian group isomorphic to Z x C4
To: Integer Ring
Arithmetic performed after coercions.
Result lives in Integer Ring
Integer Ring
"""
self._values = values
self._values_group = values_group
AbelianGroup_class.__init__(self, generator_orders, names)
self._populate_coercion_lists_(embedding=self.values_embedding())
if self.ngens() != len(self._values):
raise ValueError('need one value per generator')
def __init__(self, invariants, names, number_field, gens, proof=True):
r"""
Create a class group.
Note that the error in the test suite below is caused by the fact that
there is no category of additive abelian groups.
EXAMPLES::
sage: K.<a> = NumberField(x^2 + 23)
sage: G = K.class_group(); G
Class group of order 3 with structure C3 of Number Field in a with defining polynomial x^2 + 23
sage: G.category()
Category of groups
sage: TestSuite(G).run() # see #7945
Failure in _test_category:
...
The following tests failed: _test_elements
"""
AbelianGroup_class.__init__(self, len(invariants), invariants, names)
self._proof_flag = proof
self.__number_field = number_field
self.__gens = Sequence([FractionalIdealClass(self, x) for x in gens],
immutable=True,
universe=self,
check=False)
def __init__(self, invariants, names, number_field, gens, proof=True):
r"""
Create a class group.
Note that the error in the test suite below is caused by the fact that
there is no category of additive abelian groups.
EXAMPLES::
sage: K.<a> = NumberField(x^2 + 23)
sage: G = K.class_group(); G
Class group of order 3 with structure C3 of Number Field in a with defining polynomial x^2 + 23
sage: G.category()
Category of groups
sage: TestSuite(G).run() # see #7945
Failure in _test_category:
...
The following tests failed: _test_elements
"""
AbelianGroup_class.__init__(self, len(invariants), invariants, names)
self._proof_flag = proof
self.__number_field = number_field
self.__gens = Sequence([FractionalIdealClass(self, x) for x in gens], immutable=True,
universe=self, check=False)
def __init__(self, number_field, proof=True):
"""
Create a unit group of a number field.
INPUT:
- ``number_field`` - a number field
- ``proof`` - boolean (default True): proof flag
The proof flag is passed to pari via the ``pari_bnf()`` function
which computes the unit group. See the documentation for the
number_field module.
EXAMPLES::
sage: x = polygen(QQ)
sage: K.<a> = NumberField(x^2-38)
sage: UK = K.unit_group(); UK
Unit group with structure C2 x Z of Number Field in a with defining polynomial x^2 - 38
sage: UK.gens()
[-1, 6*a - 37]
sage: K.<a> = QuadraticField(-3)
sage: UK = K.unit_group(); UK
Unit group with structure C6 of Number Field in a with defining polynomial x^2 + 3
sage: UK.gens()
[-1/2*a + 1/2]
sage: K.<z> = CyclotomicField(13)
sage: UK = K.unit_group(); UK
Unit group with structure C26 x Z x Z x Z x Z x Z of Cyclotomic Field of order 13 and degree 12
sage: UK.gens() # random
[-z^11, z^5 + z^3, z^6 + z^5, z^9 + z^7 + z^5, z^9 + z^5 + z^4 + 1, z^5 + z]
"""
proof = get_flag(proof, "number_field")
K = number_field
pK = K.pari_bnf(proof)
self.__number_field = K
# compute the units via pari:
fu = [K(u) for u in pK.bnfunit()]
# compute a torsion generator and pick the 'simplest' one:
n, z = pK.nfrootsof1()
n = ZZ(n)
self.__ntu = n
z = K(z)
# If we replaced z by another torsion generator we would need
# to allow for this in the dlog function! So we do not.
# Store the actual generators (torsion first):
gens = [z] + fu
self.__nfu = len(fu)
self.__gens = Sequence(gens, immutable=True, universe=self, check=False)
# Construct the abtract group:
AbelianGroup_class.__init__(self, 1 + len(fu), [n] + [0] * len(fu), "u")
def __init__(self, number_field, proof=True):
"""
Create a unit group of a number field.
INPUT:
- ``number_field`` - a number field
- ``proof`` - boolean (default True): proof flag
The proof flag is passed to pari via the ``pari_bnf()`` function
which computes the unit group. See the documentation for the
number_field module.
EXAMPLES::
sage: x = polygen(QQ)
sage: K.<a> = NumberField(x^2-38)
sage: UK = K.unit_group(); UK
Unit group with structure C2 x Z of Number Field in a with defining polynomial x^2 - 38
sage: UK.gens()
[-1, 6*a - 37]
sage: K.<a> = QuadraticField(-3)
sage: UK = K.unit_group(); UK
Unit group with structure C6 of Number Field in a with defining polynomial x^2 + 3
sage: UK.gens()
[-1/2*a + 1/2]
sage: K.<z> = CyclotomicField(13)
sage: UK = K.unit_group(); UK
Unit group with structure C26 x Z x Z x Z x Z x Z of Cyclotomic Field of order 13 and degree 12
sage: UK.gens() # random
[-z^11, z^5 + z^3, z^6 + z^5, z^9 + z^7 + z^5, z^9 + z^5 + z^4 + 1, z^5 + z]
"""
proof = get_flag(proof, "number_field")
K = number_field
pK = K.pari_bnf(proof)
self.__number_field = K
# compute the units via pari:
fu = [K(u) for u in pK.bnfunit()]
# compute a torsion generator and pick the 'simplest' one:
n, z = pK.nfrootsof1()
n = ZZ(n)
self.__ntu = n
z = K(z)
# If we replaced z by another torsion generator we would need
# to allow for this in the dlog function! So we do not.
# Store the actual generators (torsion first):
gens = [z] + fu
self.__nfu = len(fu)
self.__gens = Sequence(gens, immutable=True, universe=self, check=False)
# Construct the abtract group:
AbelianGroup_class.__init__(self, 1+len(fu), [n]+[0]*len(fu), 'u')
def __init__(self,
field,
generator_orders,
algorithm=None,
gen_names='sigma'):
r"""
Initialize this Galois group.
TESTS::
sage: TestSuite(GF(9).galois_group()).run()
"""
self._field = field
self._default_algorithm = algorithm
AbelianGroup_class.__init__(self, generator_orders, gen_names)
def gen(self, i=0):
"""
The `i`-th generator of the abelian group.
INPUT:
- ``i`` -- integer (default: 0). The index of the generator.
OUTPUT:
A group element.
EXAMPLES::
sage: F = AbelianGroupWithValues([1,2,3,4,5], 5,[],names='a')
sage: F.0
a0
sage: F.0.value()
1
sage: F.2
a2
sage: F.2.value()
3
sage: G = AbelianGroupWithValues([-1,0,1], [2,1,3])
sage: G.gens()
(f0, 1, f2)
"""
g = AbelianGroup_class.gen(self, i)
g._value = self._values[i]
return g
def gen(self, i=0):
"""
The `i`-th generator of the abelian group.
INPUT:
- ``i`` -- integer (default: 0). The index of the generator.
OUTPUT:
A group element.
EXAMPLES::
sage: F = AbelianGroupWithValues([1,2,3,4,5], 5,[],names='a')
sage: F.0
a0
sage: F.0.value()
1
sage: F.2
a2
sage: F.2.value()
3
sage: G = AbelianGroupWithValues([-1,0,1], [2,1,3])
sage: G.gens()
(f0, 1, f2)
"""
g = AbelianGroup_class.gen(self, i)
g._value = self._values[i]
return g
def __init__(self, number_field, mod_ideal=1, mod_archimedean=None):
if mod_archimedean is None:
mod_archimedean = [0] * len(number_field.real_places())
mod_ideal = number_field.ideal(mod_ideal)
bnf = gp(number_field.pari_bnf())
# Use PARI to compute ray class group
bnr = bnf.bnrinit([mod_ideal, mod_archimedean], 1)
invariants = bnr[5][2] # bnr.clgp.cyc
invariants = tuple([Integer(x) for x in invariants])
names = tuple(["I%i" % i for i in range(len(invariants))])
generators = bnr[5][3] # bnr.gen = bnr.clgp[3]
generators = [number_field.ideal(pari(x)) for x in generators]
AbelianGroup_class.__init__(self, invariants, names)
self.__number_field = number_field
self.__bnr = bnr
self.__pari_mod = bnr[2][1]
self.__mod_ideal = mod_ideal
self.__mod_arch = mod_archimedean
self.__generators = generators
def __init__(self, number_field, mod_ideal = 1, mod_archimedean = None):
if mod_archimedean == None:
mod_archimedean = [0] * len(number_field.real_places())
mod_ideal = number_field.ideal( mod_ideal )
bnf = gp(number_field.pari_bnf())
# Use PARI to compute ray class group
bnr = bnf.bnrinit([mod_ideal, mod_archimedean],1)
invariants = bnr[5][2] # bnr.clgp.cyc
invariants = tuple([ Integer(x) for x in invariants ])
names = tuple([ "I%i"%i for i in range(len(invariants)) ])
generators = bnr[5][3] # bnr.gen = bnr.clgp[3]
generators = [ number_field.ideal(pari(x)) for x in generators ]
AbelianGroup_class.__init__(self, invariants, names)
self.__number_field = number_field
self.__bnr = bnr
self.__pari_mod = bnr[2][1]
self.__mod_ideal = mod_ideal
self.__mod_arch = mod_archimedean
self.__generators = generators
def __init__(self, invariants, names, number_field, gens, S, proof=True):
r"""
Create an S-class group.
EXAMPLES::
sage: K.<a> = QuadraticField(-14)
sage: I = K.ideal(2,a)
sage: S = (I,)
sage: K.S_class_group(S)
S-class group of order 2 with structure C2 of Number Field in a with defining polynomial x^2 + 14
sage: K.<a> = QuadraticField(-105)
sage: K.S_class_group([K.ideal(13, a + 8)])
S-class group of order 4 with structure C2 x C2 of Number Field in a with defining polynomial x^2 + 105
"""
AbelianGroup_class.__init__(self, len(invariants), invariants, names)
self._proof_flag = proof
self.__number_field = number_field
self.__S = S
self.__gens = Sequence([SFractionalIdealClass(self, x) for x in gens], immutable=True,
universe=self, check=False)
def __init__(self, invariants, names, number_field, gens, S, proof=True):
r"""
Create an S-class group.
EXAMPLES::
sage: K.<a> = QuadraticField(-14)
sage: I = K.ideal(2,a)
sage: S = (I,)
sage: K.S_class_group(S)
S-class group of order 2 with structure C2 of Number Field in a with defining polynomial x^2 + 14
sage: K.<a> = QuadraticField(-105)
sage: K.S_class_group([K.ideal(13, a + 8)])
S-class group of order 4 with structure C2 x C2 of Number Field in a with defining polynomial x^2 + 105
"""
AbelianGroup_class.__init__(self, len(invariants), invariants, names)
self._proof_flag = proof
self.__number_field = number_field
self.__S = S
self.__gens = Sequence([SFractionalIdealClass(self, x) for x in gens],
immutable=True,
universe=self,
check=False)