def __init__(self, p, a, b, s, t, sign):
r"""
Construct the similarity (x,y) mapsto (ax-by+s,bx+ay+t) if sign=1,
and (ax+by+s,bx-ay+t) if sign=-1
"""
if p is None:
raise ValueError("The parent must be provided")
if parent(a) is not p.base_ring():
raise ValueError("wrong parent for a")
if parent(b) is not p.base_ring():
raise ValueError("wrong parent for b")
if parent(s) is not p.base_ring():
raise ValueError("wrong parent for s")
if parent(t) is not p.base_ring():
raise ValueError("wrong parent for t")
if parent(sign) is not ZZ or not sign.is_unit():
raise ValueError("sign must be either 1 or -1.")
self._a = a
self._b = b
self._s = s
self._t = t
self._sign = sign
MultiplicativeGroupElement.__init__(self, p)
def __init__(self, F, x):
"""
Create the element x of the AbelianGroup F.
EXAMPLES::
sage: F = AbelianGroup(5, [3,4,5,8,7], 'abcde')
sage: a, b, c, d, e = F.gens()
sage: a^2 * b^3 * a^2 * b^-4
a*b^3
sage: b^-11
b
sage: a^-11
a
sage: a*b in F
True
"""
MultiplicativeGroupElement.__init__(self, F)
self.__repr = None
n = F.ngens()
if isinstance(x, (int, Integer)) and x == 1:
self.__element_vector = [0 for i in range(n)]
elif isinstance(x, list):
if len(x) != n:
raise IndexError, \
"Argument length (= %s) must be %s."%(len(x), n)
self.__element_vector = x
else:
raise TypeError, "Argument x (= %s) is of wrong type." % x
def __init__(self, F, x):
"""
Create the element x of the AbelianGroup F.
EXAMPLES::
sage: F = AbelianGroup(5, [3,4,5,8,7], 'abcde')
sage: a, b, c, d, e = F.gens()
sage: a^2 * b^3 * a^2 * b^-4
a*b^3
sage: b^-11
b
sage: a^-11
a
sage: a*b in F
True
"""
MultiplicativeGroupElement.__init__(self, F)
self.__repr = None
n = F.ngens()
if isinstance(x, (int, Integer)) and x == 1:
self.__element_vector = [ 0 for i in range(n) ]
elif isinstance(x, list):
if len(x) != n:
raise IndexError, \
"Argument length (= %s) must be %s."%(len(x), n)
self.__element_vector = x
else:
raise TypeError, "Argument x (= %s) is of wrong type."%x
def __init__(self, parent, rdict):
r"""
Create an eta product object. Usually called implicitly via
EtaGroup_class.__call__ or the EtaProduct factory function.
EXAMPLES::
sage: EtaGroupElement(EtaGroup(8), {1:24, 2:-24})
Eta product of level 8 : (eta_1)^24 (eta_2)^-24
sage: g = _; g == loads(dumps(g))
True
"""
MultiplicativeGroupElement.__init__(self, parent)
self._N = self.parent().level()
N = self._N
if isinstance(rdict, EtaGroupElement):
rdict = rdict._rdict
# Note: This is needed because the "x in G" test tries to call G(x)
# and see if it returns an error. So sometimes this will be getting
# called with rdict being an eta product, not a dictionary.
if rdict == 1:
rdict = {}
# Check Ligozat criteria
sumR = sumDR = sumNoverDr = 0
prod = 1
for d in rdict.keys():
if N % d:
raise ValueError("%s does not divide %s" % (d, N))
for d in rdict.keys():
if rdict[d] == 0:
rdict.pop(d)
continue
sumR += rdict[d]
sumDR += rdict[d] * d
sumNoverDr += rdict[d] * N / d
prod *= (N / d)**rdict[d]
if sumR != 0:
raise ValueError("sum r_d (=%s) is not 0" % sumR)
if (sumDR % 24) != 0:
raise ValueError("sum d r_d (=%s) is not 0 mod 24" % sumDR)
if (sumNoverDr % 24) != 0:
raise ValueError("sum (N/d) r_d (=%s) is not 0 mod 24" %
sumNoverDr)
if not is_square(prod):
raise ValueError("product (N/d)^(r_d) (=%s) is not a square" %
prod)
self._sumDR = sumDR # this is useful to have around
self._rdict = rdict
self._keys = rdict.keys() # avoid factoring N every time
def __init__(self, parent, polys, edge, edge_rev, check=True, reduced=False):
self._polys = tuple(polys)
self._edges = tuple(edge)
self._edges_rev = tuple(edge_rev)
MultiplicativeGroupElement.__init__(self, parent)
if not reduced:
self._reduce()
if check:
self._check()
def __init__(self, parent, a, b, s, t):
r'''Construct the similarity (x,y) mapsto (ax-by+s,bx+ay+t).'''
if parent is None:
raise ValueError("The parent must be provided")
self._a=a
self._b=b
self._s=s
self._t=t
self._parent=parent
MultiplicativeGroupElement.__init__(self,parent)
def __init__(self, parent, rdict):
r"""
Create an eta product object. Usually called implicitly via
EtaGroup_class.__call__ or the EtaProduct factory function.
EXAMPLE::
sage: EtaGroupElement(EtaGroup(8), {1:24, 2:-24})
Eta product of level 8 : (eta_1)^24 (eta_2)^-24
sage: g = _; g == loads(dumps(g))
True
"""
MultiplicativeGroupElement.__init__(self, parent)
self._N = self.parent().level()
N = self._N
if isinstance(rdict, EtaGroupElement):
rdict = rdict._rdict
# Note: This is needed because the "x in G" test tries to call G(x)
# and see if it returns an error. So sometimes this will be getting
# called with rdict being an eta product, not a dictionary.
if rdict == 1:
rdict = {}
# Check Ligozat criteria
sumR = sumDR = sumNoverDr = 0
prod = 1
for d in rdict.keys():
if N % d:
raise ValueError("%s does not divide %s" % (d, N))
for d in rdict.keys():
if rdict[d] == 0:
rdict.pop(d)
continue
sumR += rdict[d]
sumDR += rdict[d]*d
sumNoverDr += rdict[d]*N/d
prod *= (N/d)**rdict[d]
if sumR != 0:
raise ValueError("sum r_d (=%s) is not 0" % sumR)
if (sumDR % 24) != 0:
raise ValueError("sum d r_d (=%s) is not 0 mod 24" % sumDR)
if (sumNoverDr % 24) != 0:
raise ValueError("sum (N/d) r_d (=%s) is not 0 mod 24" % sumNoverDr)
if not is_square(prod):
raise ValueError("product (N/d)^(r_d) (=%s) is not a square" % prod)
self._sumDR = sumDR # this is useful to have around
self._rdict = rdict
self._keys = rdict.keys() # avoid factoring N every time
def __init__(self, parent, **kwds):
r"""
Initialize ``self``.
TESTS::
sage: L.<X,Y,Z> = LieAlgebra(QQ, 2, step=2)
sage: G = L.lie_group()
sage: g = G.exp(X)
sage: TestSuite(g).run()
"""
MultiplicativeGroupElement.__init__(self, parent)
DifferentiableManifold.Element.__init__(self, parent, **kwds)
def __init__(self, parent, **kwds):
r"""
Initialize ``self``.
TESTS::
sage: L.<X,Y,Z> = LieAlgebra(QQ, 2, step=2)
sage: G = L.lie_group()
sage: g = G.exp(X)
sage: TestSuite(g).run()
"""
MultiplicativeGroupElement.__init__(self, parent)
DifferentiableManifold.Element.__init__(self, parent, **kwds)
def __init__(self, parent, colors, perm):
"""
Initialize ``self``.
TESTS::
sage: C = ColoredPermutations(4, 3)
sage: s1,s2,t = C.gens()
sage: TestSuite(s1*s2*t).run()
"""
self._colors = tuple(colors)
self._perm = perm
MultiplicativeGroupElement.__init__(self, parent=parent)
def __init__(self, parent, colors, perm):
"""
Initialize ``self``.
TESTS::
sage: C = ColoredPermutations(4, 3)
sage: s1,s2,t = C.gens()
sage: TestSuite(s1*s2*t).run()
"""
self._colors = tuple(colors)
self._perm = perm
MultiplicativeGroupElement.__init__(self, parent=parent)
def __init__(self, parent, x):
r"""
This should not be called directly
EXAMPLES::
sage: from sage.combinat.root_system.fundamental_group import FundamentalGroupOfExtendedAffineWeylGroup
sage: x = FundamentalGroupOfExtendedAffineWeylGroup(['A',4,1], prefix="f").an_element()
sage: TestSuite(x).run()
"""
if x not in parent.special_nodes():
raise ValueError("%s is not a special node" % x)
self._value = x
MultiplicativeGroupElement.__init__(self, parent)
def __init__(self, parent, i, r, q):
if parent is None:
raise ValueError("The parent must be provided")
# I should assert that the element lives in the domain of the group.
assert i in ZZ
assert r in _Q
# Actually q should be in {1,-1,-i,i,j,-j,k,-k}. I'm not testing for that.
assert q in _Q
# There is one more condition. The group doesn't have full image...
self._i=i
self._r=r
self._q=q
self._parent=parent
MultiplicativeGroupElement.__init__(self,parent)
def __init__(self, parent, i, r, q):
if parent is None:
raise ValueError("The parent must be provided")
# I should assert that the element lives in the domain of the group.
assert i in ZZ
assert r in _Q
# Actually q should be in {1,-1,-i,i,j,-j,k,-k}. I'm not testing for that.
assert q in _Q
# There is one more condition. The group doesn't have full image...
self._i=i
self._r=r
self._q=q
self._parent=parent
MultiplicativeGroupElement.__init__(self,parent)
def __init__(self, parent, x):
r"""
This should not be called directly
EXAMPLES::
sage: from sage.combinat.root_system.fundamental_group import FundamentalGroupOfExtendedAffineWeylGroup
sage: x = FundamentalGroupOfExtendedAffineWeylGroup(['A',4,1], prefix="f").an_element()
sage: TestSuite(x).run()
"""
if x not in parent.special_nodes():
raise ValueError("%s is not a special node" % x)
self._value = x
MultiplicativeGroupElement.__init__(self, parent)
def __init__(self, parent, a, b, s, t, sign):
r'''Construct the similarity (x,y) mapsto (ax-by+s,bx+ay+t) if sign=1,
and (ax+by+s,bx-ay+t) if sign=-1
'''
if parent is None:
raise ValueError("The parent must be provided")
self._a=a
self._b=b
self._s=s
self._t=t
if not (sign==1 or sign==-1):
raise ValueError("sign must be either 1 or -1.")
self._sign=sign
self._parent=parent
MultiplicativeGroupElement.__init__(self,parent)
def __init__(self,
parent,
polys,
edge,
edge_rev,
check=True,
reduced=False):
self._polys = tuple(polys)
self._edges = tuple(edge)
self._edges_rev = tuple(edge_rev)
MultiplicativeGroupElement.__init__(self, parent)
if not reduced:
self._reduce()
if check:
self._check()
def __init__(self,
parent,
word_rep=None,
quaternion_rep=None,
check=False):
r'''
Initialize.
INPUT:
- a list of the form [(g1,a1),(g2,a2),...,(gn,an)] where the gi are indices
refering to fixed generators, and the ai are integers, or
an element of the quaternion algebra ``self.parent().quaternion_algebra()``.
'''
MultiplicativeGroupElement.__init__(self, parent)
init_data = False
self.has_word_rep = False
self.has_quaternion_rep = False
if word_rep is not None:
self.word_rep = word_rep
self.has_word_rep = True
init_data = True
if quaternion_rep is not None:
if check:
if not parent._is_in_order(quaternion_rep):
raise ValueError('Quaternion (= %s) must be in order' %
quaternion_rep)
if parent.base_field() == QQ:
rednrm = quaternion_rep.reduced_norm() if self.parent(
).discriminant != 1 else quaternion_rep.determinant()
if rednrm != 1:
raise ValueError('Quaternion must be of norm 1')
else:
rednrm = quaternion_rep.reduced_norm()
if not rednrm.is_integral() or not (1 /
rednrm).is_integral():
raise ValueError('Quaternion must be of unit norm')
if self.has_word_rep:
self.check_consistency(quaternion_rep, self.word_rep)
self.quaternion_rep = quaternion_rep
set_immutable(self.quaternion_rep)
self.has_quaternion_rep = True
init_data = True
if init_data is False:
raise ValueError('Must pass either quaternion_rep or word_rep')
def __init__(self, parent, *args):
r'''
Construct the translation (x,y) mapsto (x+s,y+t).
Arguments:
The first argument must be the parent.
If there is one additional argument (call it a) then s=a[0] and t=a[1].
If there are two elements, these perscribe s and t.
'''
if parent is None:
raise ValueError("The parent must be provided")
if len(args)==1:
self._s=args[0][0]
self._t=args[0][1]
if len(args)==2:
self._s=args[0]
self._t=args[1]
self._parent=parent
MultiplicativeGroupElement.__init__(self,parent)
def __init__(self, parent, exponents):
"""
Create an element.
EXAMPLES::
sage: F = AbelianGroup(3,[7,8,9])
sage: Fd = F.dual_group(names="ABC")
sage: A,B,C = Fd.gens()
sage: A*B^-1 in Fd
True
"""
MultiplicativeGroupElement.__init__(self, parent)
n = parent.ngens()
if exponents == 1:
self._exponents = tuple( ZZ.zero() for i in range(n) )
else:
self._exponents = tuple( ZZ(e) for e in exponents )
if len(self._exponents) != n:
raise IndexError('argument length (= %s) must be %s.'%(len(exponents), n))
def __init__(self, p, a, b, s, t, sign):
r"""
Construct the similarity (x,y) mapsto (ax-by+s,bx+ay+t) if sign=1,
and (ax+by+s,bx-ay+t) if sign=-1
"""
if p is None:
raise ValueError("The parent must be provided")
field = p._field
if parent(a) is not field or \
parent(b) is not field or \
parent(s) is not field or \
parent(t) is not field:
raise ValueError("wrong parent for a,b,s or t")
self._a = a
self._b = b
self._s = s
self._t = t
if parent(sign) is not ZZ or not sign.is_unit():
raise ValueError("sign must be either 1 or -1.")
self._sign = sign
MultiplicativeGroupElement.__init__(self, p)
def __init__(self, parent, exponents):
"""
Create an element.
EXAMPLES::
sage: F = AbelianGroup(3,[7,8,9])
sage: Fd = F.dual_group(names="ABC")
sage: A,B,C = Fd.gens()
sage: A*B^-1 in Fd
True
"""
MultiplicativeGroupElement.__init__(self, parent)
n = parent.ngens()
if exponents == 1:
self._exponents = tuple(ZZ.zero() for i in range(n))
else:
self._exponents = tuple(ZZ(e) for e in exponents)
if len(self._exponents) != n:
raise IndexError('argument length (= %s) must be %s.' %
(len(exponents), n))
def __init__(self, parent, x, check=True):
"""
Create an element of an arithmetic subgroup.
INPUT:
- parent - an arithmetic subgroup
- x - data defining a 2x2 matrix over ZZ
which lives in parent
- check - if True, check that parent
is an arithmetic subgroup, and that
x defines a matrix of determinant 1
We tend not to create elements of arithmetic subgroups that aren't
SL2Z, in order to avoid coercion issues (that is, the other arithmetic
subgroups are "facade parents").
EXAMPLES::
sage: G = Gamma0(27)
sage: sage.modular.arithgroup.arithgroup_element.ArithmeticSubgroupElement(G, [4,1,27,7])
[ 4 1]
[27 7]
sage: sage.modular.arithgroup.arithgroup_element.ArithmeticSubgroupElement(Integers(3), [4,1,27,7])
Traceback (most recent call last):
...
TypeError: parent (= Ring of integers modulo 3) must be an arithmetic subgroup
sage: sage.modular.arithgroup.arithgroup_element.ArithmeticSubgroupElement(G, [2,0,0,2])
Traceback (most recent call last):
...
TypeError: matrix must have determinant 1
sage: sage.modular.arithgroup.arithgroup_element.ArithmeticSubgroupElement(G, [2,0,0,2], check=False)
[2, 0, 0, 2]
sage: x = Gamma0(11)([2,1,11,6])
sage: x == loads(dumps(x))
True
sage: x = Gamma0(5).0
sage: SL2Z(x)
[1 1]
[0 1]
sage: x in SL2Z
True
"""
if check:
if not is_ArithmeticSubgroup(parent):
raise TypeError, "parent (= %s) must be an arithmetic subgroup"%parent
x = mi2x2(M2Z, x, copy=True, coerce=True)
if x.determinant() != 1:
raise TypeError, "matrix must have determinant 1"
try:
x.set_immutable()
except AttributeError:
pass
MultiplicativeGroupElement.__init__(self, parent)
self.__x = x
def __init__(self, parent, x, check=True):
"""
Create an element of an arithmetic subgroup.
INPUT:
- parent - an arithmetic subgroup
- x - data defining a 2x2 matrix over ZZ
which lives in parent
- check - if True, check that parent
is an arithmetic subgroup, and that
x defines a matrix of determinant 1
We tend not to create elements of arithmetic subgroups that aren't
SL2Z, in order to avoid coercion issues (that is, the other arithmetic
subgroups are "facade parents").
EXAMPLES::
sage: G = Gamma0(27)
sage: sage.modular.arithgroup.arithgroup_element.ArithmeticSubgroupElement(G, [4,1,27,7])
[ 4 1]
[27 7]
sage: sage.modular.arithgroup.arithgroup_element.ArithmeticSubgroupElement(Integers(3), [4,1,27,7])
Traceback (most recent call last):
...
TypeError: parent (= Ring of integers modulo 3) must be an arithmetic subgroup
sage: sage.modular.arithgroup.arithgroup_element.ArithmeticSubgroupElement(G, [2,0,0,2])
Traceback (most recent call last):
...
TypeError: matrix must have determinant 1
sage: sage.modular.arithgroup.arithgroup_element.ArithmeticSubgroupElement(G, [2,0,0,2], check=False)
[2, 0, 0, 2]
sage: x = Gamma0(11)([2,1,11,6])
sage: x == loads(dumps(x))
True
sage: x = Gamma0(5).0
sage: SL2Z(x)
[1 1]
[0 1]
sage: x in SL2Z
True
"""
if check:
if not is_ArithmeticSubgroup(parent):
raise TypeError, "parent (= %s) must be an arithmetic subgroup" % parent
x = mi2x2(M2Z, x, copy=True, coerce=True)
if x.determinant() != 1:
raise TypeError, "matrix must have determinant 1"
try:
x.set_immutable()
except AttributeError:
pass
MultiplicativeGroupElement.__init__(self, parent)
self.__x = x
