def __init__(self, gens_orders, names, number_field, gens, proof=True):
r"""
Create a class group.
TESTS::
sage: K.<a> = NumberField(x^2 + 23)
sage: G = K.class_group()
sage: TestSuite(G).run()
"""
AbelianGroupWithValues_class.__init__(self, gens_orders, names, gens,
values_group=number_field.ideal_monoid())
self._proof_flag = proof
self._number_field = number_field
def __init__(self, gens_orders, names, number_field, gens, proof=True):
r"""
Create a class group.
TESTS::
sage: K.<a> = NumberField(x^2 + 23)
sage: G = K.class_group()
sage: TestSuite(G).run()
"""
AbelianGroupWithValues_class.__init__(self, gens_orders, names, gens,
values_group=number_field.ideal_monoid())
self._proof_flag = proof
self._number_field = number_field
def __init__(self, gens_orders, 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 with a = 3.741657386773942?*I
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 with a = 10.24695076595960?*I
"""
AbelianGroupWithValues_class.__init__(self, gens_orders, names, gens,
values_group=number_field.ideal_monoid())
self._proof_flag = proof
self._number_field = number_field
self._S = S
def __init__(self, gens_orders, 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
"""
AbelianGroupWithValues_class.__init__(self, gens_orders, names, gens,
values_group=number_field.ideal_monoid())
self._proof_flag = proof
self._number_field = number_field
self._S = S
def __init__(self, number_field, proof=True, S=None):
"""
Create a unit group of a number field.
INPUT:
- ``number_field`` - a number field
- ``proof`` - boolean (default True): proof flag
- ``S`` - tuple of prime ideals, or an ideal, or a single
ideal or element from which an ideal can be constructed, in
which case the support is used. If None, the global unit
group is constructed; otherwise, the S-unit group is
constructed.
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()
(u0, u1)
sage: UK.gens_values()
[-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()
(u,)
sage: UK.gens_values()
[-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()
(u0, u1, u2, u3, u4, u5)
sage: UK.gens_values() # 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]
sage: SUK = UnitGroup(K,S=2); SUK
S-unit group with structure C26 x Z x Z x Z x Z x Z x Z of Cyclotomic Field of order 13 and degree 12 with S = (Fractional ideal (2),)
"""
proof = get_flag(proof, "number_field")
K = number_field
pK = K.pari_bnf(proof)
self.__number_field = K
self.__pari_number_field = pK
# process the parameter S:
if not S:
S = self.__S = ()
else:
if isinstance(S, list):
S = tuple(S)
if not isinstance(S, tuple):
try:
S = tuple(K.ideal(S).prime_factors())
except (NameError, TypeError, ValueError):
raise ValueError("Cannot make a set of primes from %s" %
(S, ))
else:
try:
S = tuple(K.ideal(P) for P in S)
except (NameError, TypeError, ValueError):
raise ValueError("Cannot make a set of primes from %s" %
(S, ))
if not all([P.is_prime() for P in S]):
raise ValueError(
"Not all elements of %s are prime ideals" % (S, ))
self.__S = S
self.__pS = pS = [P.pari_prime() for P in S]
# compute the fundamental units via pari:
fu = [K(u) for u in pK.bnfunit()]
self.__nfu = len(fu)
# compute the additional S-unit generators:
if S:
self.__S_unit_data = pK.bnfsunit(pS)
su = [K(u) for u in self.__S_unit_data[0]]
else:
su = []
self.__nsu = len(su)
self.__rank = self.__nfu + self.__nsu
# 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 + su
values = Sequence(gens, immutable=True, universe=self, check=False)
# Construct the abtract group:
gens_orders = tuple([ZZ(n)] + [ZZ(0)] * (self.__rank))
AbelianGroupWithValues_class.__init__(self, gens_orders, 'u', values,
number_field)
def __init__(self, number_field, proof=True, S=None):
"""
Create a unit group of a number field.
INPUT:
- ``number_field`` - a number field
- ``proof`` - boolean (default True): proof flag
- ``S`` - tuple of prime ideals, or an ideal, or a single
ideal or element from which an ideal can be constructed, in
which case the support is used. If None, the global unit
group is constructed; otherwise, the S-unit group is
constructed.
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()
(u0, u1)
sage: UK.gens_values()
[-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 with a = 1.732050807568878?*I
sage: UK.gens()
(u,)
sage: UK.gens_values()
[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()
(u0, u1, u2, u3, u4, u5)
sage: UK.gens_values() # 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]
sage: SUK = UnitGroup(K,S=2); SUK
S-unit group with structure C26 x Z x Z x Z x Z x Z x Z of Cyclotomic Field of order 13 and degree 12 with S = (Fractional ideal (2),)
TESTS:
Number fields defined by non-monic and non-integral
polynomials are supported (:trac:`252`);
the representation depends on the PARI version::
sage: K.<a> = NumberField(7/9*x^3 + 7/3*x^2 - 56*x + 123)
sage: K.unit_group()
Unit group with structure C2 x Z x Z of Number Field in a with defining polynomial 7/9*x^3 + 7/3*x^2 - 56*x + 123
sage: UnitGroup(K, S=tuple(K.primes_above(7)))
S-unit group with structure C2 x Z x Z x Z of Number Field in a with defining polynomial 7/9*x^3 + 7/3*x^2 - 56*x + 123 with S = (Fractional ideal (...),)
sage: K.primes_above(7)[0] in (7/225*a^2 - 7/75*a - 42/25, 28/225*a^2 + 77/75*a - 133/25)
True
Conversion from unit group to a number field and back
gives the right results (:trac:`25874`)::
sage: K = QuadraticField(-3).composite_fields(QuadraticField(2))[0]
sage: U = K.unit_group()
sage: tuple(U(K(u)) for u in U.gens()) == U.gens()
True
sage: US = K.S_unit_group(3)
sage: tuple(US(K(u)) for u in US.gens()) == US.gens()
True
"""
proof = get_flag(proof, "number_field")
K = number_field
pK = K.pari_bnf(proof)
self.__number_field = K
self.__pari_number_field = pK
# process the parameter S:
if not S:
S = self.__S = ()
else:
if isinstance(S, list):
S = tuple(S)
if not isinstance(S, tuple):
try:
S = tuple(K.ideal(S).prime_factors())
except (NameError, TypeError, ValueError):
raise ValueError("Cannot make a set of primes from %s" %
(S, ))
else:
try:
S = tuple(K.ideal(P) for P in S)
except (NameError, TypeError, ValueError):
raise ValueError("Cannot make a set of primes from %s" %
(S, ))
if not all(P.is_prime() for P in S):
raise ValueError(
"Not all elements of %s are prime ideals" % (S, ))
self.__S = S
self.__pS = pS = [P.pari_prime() for P in S]
# compute the fundamental units via pari:
fu = [K(u, check=False) for u in pK.bnfunit()]
self.__nfu = len(fu)
# compute the additional S-unit generators:
if S:
self.__S_unit_data = pK.bnfsunit(pS)
su = [K(u, check=False) for u in self.__S_unit_data[0]]
else:
su = []
self.__nsu = len(su)
self.__rank = self.__nfu + self.__nsu
# compute a torsion generator and pick the 'simplest' one:
n, z = pK[7][
3] # number of roots of unity and bnf.tu as in pari documentation
n = ZZ(n)
self.__ntu = n
z = K(z, check=False)
# 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 + su
values = Sequence(gens, immutable=True, universe=self, check=False)
# Construct the abtract group:
gens_orders = tuple([ZZ(n)] + [ZZ(0)] * (self.__rank))
AbelianGroupWithValues_class.__init__(self, gens_orders, 'u', values,
number_field)
def __init__(self, number_field, proof=True, S=None):
"""
Create a unit group of a number field.
INPUT:
- ``number_field`` - a number field
- ``proof`` - boolean (default True): proof flag
- ``S`` - tuple of prime ideals, or an ideal, or a single
ideal or element from which an ideal can be constructed, in
which case the support is used. If None, the global unit
group is constructed; otherwise, the S-unit group is
constructed.
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()
(u0, u1)
sage: UK.gens_values()
[-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()
(u,)
sage: UK.gens_values()
[-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()
(u0, u1, u2, u3, u4, u5)
sage: UK.gens_values() # 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]
sage: SUK = UnitGroup(K,S=2); SUK
S-unit group with structure C26 x Z x Z x Z x Z x Z x Z of Cyclotomic Field of order 13 and degree 12 with S = (Fractional ideal (2),)
"""
proof = get_flag(proof, "number_field")
K = number_field
pK = K.pari_bnf(proof)
self.__number_field = K
self.__pari_number_field = pK
# process the parameter S:
if not S:
S = self.__S = ()
else:
if type(S)==list:
S = tuple(S)
if not type(S)==tuple:
try:
S = tuple(K.ideal(S).prime_factors())
except (NameError, TypeError, ValueError):
raise ValueError("Cannot make a set of primes from %s"%(S,))
else:
try:
S = tuple(K.ideal(P) for P in S)
except (NameError, TypeError, ValueError):
raise ValueError("Cannot make a set of primes from %s"%(S,))
if not all([P.is_prime() for P in S]):
raise ValueError("Not all elements of %s are prime ideals"%(S,))
self.__S = S
self.__pS = pS = [P.pari_prime() for P in S]
# compute the fundamental units via pari:
fu = [K(u) for u in pK.bnfunit()]
self.__nfu = len(fu)
# compute the additional S-unit generators:
if S:
self.__S_unit_data = pK.bnfsunit(pS)
su = [K(u) for u in self.__S_unit_data[0]]
else:
su = []
self.__nsu = len(su)
self.__rank = self.__nfu + self.__nsu
# 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 + su
values = Sequence(gens, immutable=True, universe=self, check=False)
# Construct the abtract group:
gens_orders = tuple([ZZ(n)]+[ZZ(0)]*(self.__rank))
AbelianGroupWithValues_class.__init__(self, gens_orders, 'u', values, number_field)
def __init__(self, number_field, proof=True, S=None):
"""
Create a unit group of a number field.
INPUT:
- ``number_field`` - a number field
- ``proof`` - boolean (default True): proof flag
- ``S`` - tuple of prime ideals, or an ideal, or a single
ideal or element from which an ideal can be constructed, in
which case the support is used. If None, the global unit
group is constructed; otherwise, the S-unit group is
constructed.
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()
(u0, u1)
sage: UK.gens_values()
[-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 with a = 1.732050807568878?*I
sage: UK.gens()
(u,)
sage: UK.gens_values()
[-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()
(u0, u1, u2, u3, u4, u5)
sage: UK.gens_values() # 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]
sage: SUK = UnitGroup(K,S=2); SUK
S-unit group with structure C26 x Z x Z x Z x Z x Z x Z of Cyclotomic Field of order 13 and degree 12 with S = (Fractional ideal (2),)
TESTS:
Number fields defined by non-monic and non-integral
polynomials are supported (:trac:`252`);
the representation depends on the PARI version::
sage: K.<a> = NumberField(7/9*x^3 + 7/3*x^2 - 56*x + 123)
sage: K.unit_group()
Unit group with structure C2 x Z x Z of Number Field in a with defining polynomial 7/9*x^3 + 7/3*x^2 - 56*x + 123
sage: UnitGroup(K, S=tuple(K.primes_above(7)))
S-unit group with structure C2 x Z x Z x Z of Number Field in a with defining polynomial 7/9*x^3 + 7/3*x^2 - 56*x + 123 with S = (Fractional ideal (...),)
sage: K.primes_above(7)[0] in (7/225*a^2 - 7/75*a - 42/25, 28/225*a^2 + 77/75*a - 133/25)
True
Conversion from unit group to a number field and back
gives the right results (:trac:`25874`)::
sage: K = QuadraticField(-3).composite_fields(QuadraticField(2))[0]
sage: U = K.unit_group()
sage: tuple(U(K(u)) for u in U.gens()) == U.gens()
True
sage: US = K.S_unit_group(3)
sage: tuple(US(K(u)) for u in US.gens()) == US.gens()
True
"""
proof = get_flag(proof, "number_field")
K = number_field
pK = K.pari_bnf(proof)
self.__number_field = K
self.__pari_number_field = pK
# process the parameter S:
if not S:
S = self.__S = ()
else:
if isinstance(S, list):
S = tuple(S)
if not isinstance(S, tuple):
try:
S = tuple(K.ideal(S).prime_factors())
except (NameError, TypeError, ValueError):
raise ValueError("Cannot make a set of primes from %s" %
(S, ))
else:
try:
S = tuple(K.ideal(P) for P in S)
except (NameError, TypeError, ValueError):
raise ValueError("Cannot make a set of primes from %s" %
(S, ))
if not all(P.is_prime() for P in S):
raise ValueError(
"Not all elements of %s are prime ideals" % (S, ))
self.__S = S
self.__pS = pS = [P.pari_prime() for P in S]
# compute the fundamental units via pari:
fu = [K(u, check=False) for u in pK.bnf_get_fu()]
self.__nfu = len(fu)
# compute the additional S-unit generators:
if S:
self.__S_unit_data = pK.bnfunits(pS)
else:
self.__S_unit_data = pK.bnfunits()
# TODO: converting the factored matrix representation of bnfunits into polynomial
# form is a *big* waste of time
su_fu_tu = [
pK.nfbasistoalg(pK.nffactorback(z)) for z in self.__S_unit_data[0]
]
self.__nfu = len(pK.bnf_get_fu()) # number of fundamental units
self.__nsu = len(su_fu_tu) - self.__nfu - 1 # number of S-units
self.__ntu = pK.bnf_get_tu()[0] # order of torsion
self.__rank = self.__nfu + self.__nsu
# Move the torsion unit first, then fundamental units then S-units
gens = [K(u, check=False) for u in su_fu_tu]
gens = [gens[-1]] + gens[self.__nsu:-1] + gens[:self.__nsu]
# Construct the abstract group:
gens_orders = tuple([ZZ(self.__ntu)] + [ZZ(0)] * (self.__rank))
AbelianGroupWithValues_class.__init__(self, gens_orders, 'u', gens,
number_field)
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()
(u0, u1)
sage: UK.gens_values()
[-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()
(u,)
sage: UK.gens_values()
[-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()
(u0, u1, u2, u3, u4, u5)
sage: UK.gens_values() # 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
values = Sequence(gens, immutable=True, universe=self, check=False)
# Construct the abtract group:
gens_orders = tuple([ZZ(n)] + [ZZ(0)] * len(fu))
AbelianGroupWithValues_class.__init__(self, gens_orders, 'u', values,
number_field)
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()
(u0, u1)
sage: UK.gens_values()
[-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()
(u,)
sage: UK.gens_values()
[-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()
(u0, u1, u2, u3, u4, u5)
sage: UK.gens_values() # 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
values = Sequence(gens, immutable=True, universe=self, check=False)
# Construct the abtract group:
gens_orders = tuple([ZZ(n)]+[ZZ(0)]*len(fu))
AbelianGroupWithValues_class.__init__(self, gens_orders, 'u', values, number_field)
