def is_locally_solvable(self, p):
r"""
Returns True if and only if ``self`` has a solution over the
`p`-adic numbers. Here `p` is a prime number or equals
`-1`, infinity, or `\RR` to denote the infinite place.
EXAMPLES::
sage: C = Conic(QQ, [1,2,3])
sage: C.is_locally_solvable(-1)
False
sage: C.is_locally_solvable(2)
False
sage: C.is_locally_solvable(3)
True
sage: C.is_locally_solvable(QQ.hom(RR))
False
sage: D = Conic(QQ, [1, 2, -3])
sage: D.is_locally_solvable(infinity)
True
sage: D.is_locally_solvable(RR)
True
"""
from sage.categories.map import Map
from sage.categories.all import Rings
D, T = self.diagonal_matrix()
abc = [D[j, j] for j in range(3)]
if abc[2] == 0:
return True
a = -abc[0]/abc[2]
b = -abc[1]/abc[2]
if is_RealField(p) or is_InfinityElement(p):
p = -1
elif isinstance(p, Map) and p.category_for().is_subcategory(Rings()):
# p is a morphism of Rings
if p.domain() is QQ and is_RealField(p.codomain()):
p = -1
else:
raise TypeError("p (=%s) needs to be a prime of base field " \
"B ( =`QQ`) in is_locally_solvable" % p)
if hilbert_symbol(a, b, p) == -1:
if self._local_obstruction is None:
self._local_obstruction = p
return False
return True
def is_locally_solvable(self, p):
r"""
Returns True if and only if ``self`` has a solution over the
`p`-adic numbers. Here `p` is a prime number or equals
`-1`, infinity, or `\RR` to denote the infinite place.
EXAMPLES::
sage: C = Conic(QQ, [1,2,3])
sage: C.is_locally_solvable(-1)
False
sage: C.is_locally_solvable(2)
False
sage: C.is_locally_solvable(3)
True
sage: C.is_locally_solvable(QQ.hom(RR))
False
sage: D = Conic(QQ, [1, 2, -3])
sage: D.is_locally_solvable(infinity)
True
sage: D.is_locally_solvable(RR)
True
"""
from sage.categories.map import Map
from sage.categories.all import Rings
D, T = self.diagonal_matrix()
abc = [D[j, j] for j in range(3)]
if abc[2] == 0:
return True
a = -abc[0] / abc[2]
b = -abc[1] / abc[2]
if is_RealField(p) or is_InfinityElement(p):
p = -1
elif isinstance(p, Map) and p.category_for().is_subcategory(Rings()):
# p is a morphism of Rings
if p.domain() is QQ and is_RealField(p.codomain()):
p = -1
else:
raise TypeError("p (=%s) needs to be a prime of base field " \
"B ( =`QQ`) in is_locally_solvable" % p)
if hilbert_symbol(a, b, p) == -1:
if self._local_obstruction is None:
self._local_obstruction = p
return False
return True
def __init__(self, number_field, a, b=None, parent=None, lreps=None):
"""
Constructor of number field cusps. See ``NFCusp`` for full
documentation.
EXAMPLES::
sage: k.<a> = NumberField(x^2 + 1)
sage: c = NFCusp(k, 3, a+1); c
Cusp [3: a + 1] of Number Field in a with defining polynomial x^2 + 1
sage: c.parent()
Set of all cusps of Number Field in a with defining polynomial x^2 + 1
sage: kCusps = NFCusps(k)
sage: c.parent() is kCusps
True
"""
if parent is None:
parent = NFCusps(number_field)
Element.__init__(self, parent)
R = number_field.maximal_order()
if b is None:
if not a:#that is cusp "0"
self.__a = R(0)
self.__b = R(1)
return
if isinstance(a, NFCusp):
if a.parent() == parent:
self.__a = R(a.__a)
self.__b = R(a.__b)
else:
raise ValueError("Cannot coerce cusps from one field to another")
elif a in R:
self.__a = R(a)
self.__b = R(1)
elif a in number_field:
self.__b = R(a.denominator())
self.__a = R(a * self.__b)
elif is_InfinityElement(a):
self.__a = R(1)
self.__b = R(0)
elif isinstance(a, (int, long)):
self.__a = R(a)
self.__b = R(1)
elif isinstance(a, (tuple, list)):
if len(a) != 2:
raise TypeError("Unable to convert %s to a cusp \
of the number field"%a)
if a[1].is_zero():
self.__a = R(1)
self.__b = R(0)
elif a[0] in R and a[1] in R:
self.__a = R(a[0])
self.__b = R(a[1])
elif isinstance(a[0], NFCusp):#we know that a[1] is not zero
if a[1] == 1:
self.__a = a[0].__a
self.__b = a[0].__b
else:
r = a[0].__a / (a[0].__b * a[1])
self.__b = R(r.denominator())
self.__a = R(r*self.__b)
else:
try:
r = number_field(a[0]/a[1])
self.__b = R(r.denominator())
self.__a = R(r * self.__b)
except (ValueError, TypeError):
raise TypeError("Unable to convert %s to a cusp \
of the number field"%a)
else:
try:
r = number_field(a)
self.__b = R(r.denominator())
self.__a = R(r * self.__b)
except (ValueError, TypeError):
raise TypeError("Unable to convert %s to a cusp \
of the number field"%a)
else:#'b' is given
if is_InfinityElement(b):
if is_InfinityElement(a) or (isinstance(a, NFCusp) and a.is_infinity()):
raise TypeError("Unable to convert (%s, %s) \
to a cusp of the number field"%(a, b))
self.__a = R(0)
self.__b = R(1)
return
elif not b:
if not a:
raise TypeError("Unable to convert (%s, %s) \
to a cusp of the number field"%(a, b))
self.__a = R(1)
self.__b = R(0)
return
if not a:
self.__a = R(0)
self.__b = R(1)
return
if (b in R or isinstance(b, (int, long))) and (a in R or isinstance(a, (int, long))):
self.__a = R(a)
self.__b = R(b)
else:
if a in R or a in number_field:
r = a / b
elif is_InfinityElement(a):
self.__a = R(1)
self.__b = R(0)
return
elif isinstance(a, NFCusp):
if a.is_infinity():
self.__a = R(1)
self.__b = R(0)
return
r = a.__a / (a.__b * b)
elif isinstance(a, (int, long)):
r = R(a) / b
elif isinstance(a, (tuple, list)):
if len(a) != 2:
raise TypeError("Unable to convert (%s, %s) \
to a cusp of the number field"%(a, b))
r = R(a[0]) / (R(a[1]) * b)
else:
try:
r = number_field(a) / b
except (ValueError, TypeError):
raise TypeError("Unable to convert (%s, %s) \
to a cusp of the number field"%(a, b))
self.__b = R(r.denominator())
self.__a = R(r * self.__b)
if not lreps is None:
# Changes the representative of the cusp so the ideal associated
# to the cusp is one of the ideals of the given list lreps.
# Note: the trivial class is always represented by (1).
I = self.ideal()
for J in lreps:
if (J/I).is_principal():
newI = J
l = (newI/I).gens_reduced()[0]
self.__a = R(l * self.__a)
self.__b = R(l * self.__b)
def __init__(self, number_field, a, b=None, parent=None, lreps=None):
"""
Constructor of number field cusps. See ``NFCusp`` for full
documentation.
EXAMPLES::
sage: k.<a> = NumberField(x^2 + 1)
sage: c = NFCusp(k, 3, a+1); c
Cusp [3: a + 1] of Number Field in a with defining polynomial x^2 + 1
sage: c.parent()
Set of all cusps of Number Field in a with defining polynomial x^2 + 1
sage: kCusps = NFCusps(k)
sage: c.parent() is kCusps
True
"""
if parent is None:
parent = NFCusps(number_field)
Element.__init__(self, parent)
R = number_field.maximal_order()
if b is None:
if not a: #that is cusp "0"
self.__a = R(0)
self.__b = R(1)
return
if isinstance(a, NFCusp):
if a.parent() == parent:
self.__a = R(a.__a)
self.__b = R(a.__b)
else:
raise ValueError(
"Cannot coerce cusps from one field to another")
elif a in R:
self.__a = R(a)
self.__b = R(1)
elif a in number_field:
self.__b = R(a.denominator())
self.__a = R(a * self.__b)
elif is_InfinityElement(a):
self.__a = R(1)
self.__b = R(0)
elif isinstance(a, (int, long)):
self.__a = R(a)
self.__b = R(1)
elif isinstance(a, (tuple, list)):
if len(a) != 2:
raise TypeError("Unable to convert %s to a cusp \
of the number field" % a)
if a[1].is_zero():
self.__a = R(1)
self.__b = R(0)
elif a[0] in R and a[1] in R:
self.__a = R(a[0])
self.__b = R(a[1])
elif isinstance(a[0], NFCusp): #we know that a[1] is not zero
if a[1] == 1:
self.__a = a[0].__a
self.__b = a[0].__b
else:
r = a[0].__a / (a[0].__b * a[1])
self.__b = R(r.denominator())
self.__a = R(r * self.__b)
else:
try:
r = number_field(a[0] / a[1])
self.__b = R(r.denominator())
self.__a = R(r * self.__b)
except (ValueError, TypeError):
raise TypeError("Unable to convert %s to a cusp \
of the number field" % a)
else:
try:
r = number_field(a)
self.__b = R(r.denominator())
self.__a = R(r * self.__b)
except (ValueError, TypeError):
raise TypeError("Unable to convert %s to a cusp \
of the number field" % a)
else: #'b' is given
if is_InfinityElement(b):
if is_InfinityElement(a) or (isinstance(a, NFCusp)
and a.is_infinity()):
raise TypeError("Unable to convert (%s, %s) \
to a cusp of the number field" % (a, b))
self.__a = R(0)
self.__b = R(1)
return
elif not b:
if not a:
raise TypeError("Unable to convert (%s, %s) \
to a cusp of the number field" % (a, b))
self.__a = R(1)
self.__b = R(0)
return
if not a:
self.__a = R(0)
self.__b = R(1)
return
if (b in R or isinstance(b,
(int, long))) and (a in R or isinstance(
a, (int, long))):
self.__a = R(a)
self.__b = R(b)
else:
if a in R or a in number_field:
r = a / b
elif is_InfinityElement(a):
self.__a = R(1)
self.__b = R(0)
return
elif isinstance(a, NFCusp):
if a.is_infinity():
self.__a = R(1)
self.__b = R(0)
return
r = a.__a / (a.__b * b)
elif isinstance(a, (int, long)):
r = R(a) / b
elif isinstance(a, (tuple, list)):
if len(a) != 2:
raise TypeError("Unable to convert (%s, %s) \
to a cusp of the number field" %
(a, b))
r = R(a[0]) / (R(a[1]) * b)
else:
try:
r = number_field(a) / b
except (ValueError, TypeError):
raise TypeError("Unable to convert (%s, %s) \
to a cusp of the number field" %
(a, b))
self.__b = R(r.denominator())
self.__a = R(r * self.__b)
if not lreps is None:
# Changes the representative of the cusp so the ideal associated
# to the cusp is one of the ideals of the given list lreps.
# Note: the trivial class is always represented by (1).
I = self.ideal()
for J in lreps:
if (J / I).is_principal():
newI = J
l = (newI / I).gens_reduced()[0]
self.__a = R(l * self.__a)
self.__b = R(l * self.__b)
def _cmp_generators(self, other):
"""
Implement comparison
We treat two matrix groups as equal if their generators are
the same in the same order. Infinitely-generated groups are
compared by identity.
INPUT:
- ``other`` -- anything.
OUTPUT:
``-1``, ``0``, or ``+1``.
EXAMPLES::
sage: G = GL(2,3)
sage: H = MatrixGroup(G.gens())
sage: G._cmp_generators(H)
0
sage: cmp(G,H)
0
sage: cmp(H,G)
0
sage: G == H
True
"""
if not is_MatrixGroup(other):
return cmp(type(self), type(other))
c = cmp(self.matrix_space(), other.matrix_space())
if c != 0:
return c
def identity_cmp():
return cmp(id(self), id(other))
# compare number of generators
try:
n_self = self.ngens()
n_other = other.ngens()
except (AttributeError, NotImplementedError):
return identity_cmp()
c = cmp(n_self, n_other)
if c != 0:
return c
from sage.structure.element import is_InfinityElement
if is_InfinityElement(n_self) or is_InfinityElement(n_other):
return identity_cmp()
# compacte generator matrices
try:
self_gens = self.gens()
other_gens = other.gens()
except (AttributeError, NotImplementedError):
return identity_cmp()
assert(n_self == n_other)
for g,h in zip(self_gens, other_gens):
c = cmp(g.matrix(), h.matrix())
if c != 0:
return c
return c
def _cmp_generators(self, other):
"""
Implement comparison
We treat two matrix groups as equal if their generators are
the same in the same order. Infinitely-generated groups are
compared by identity.
INPUT:
- ``other`` -- anything.
OUTPUT:
``-1``, ``0``, or ``+1``.
EXAMPLES::
sage: G = GL(2,3)
sage: H = MatrixGroup(G.gens())
sage: G._cmp_generators(H)
0
sage: cmp(G,H)
0
sage: cmp(H,G)
0
sage: G == H
True
"""
if not is_MatrixGroup(other):
return cmp(type(self), type(other))
c = cmp(self.matrix_space(), other.matrix_space())
if c != 0:
return c
def identity_cmp():
return cmp(id(self), id(other))
# compare number of generators
try:
n_self = self.ngens()
n_other = other.ngens()
except (AttributeError, NotImplementedError):
return identity_cmp()
c = cmp(n_self, n_other)
if c != 0:
return c
from sage.structure.element import is_InfinityElement
if is_InfinityElement(n_self) or is_InfinityElement(n_other):
return identity_cmp()
# compacte generator matrices
try:
self_gens = self.gens()
other_gens = other.gens()
except (AttributeError, NotImplementedError):
return identity_cmp()
assert(n_self == n_other)
for g,h in zip(self_gens, other_gens):
c = cmp(g.matrix(), h.matrix())
if c != 0:
return c
return c
def __init__(self, a, b=None, parent=None, check=True):
r"""
Create the cusp a/b in `\mathbb{P}^1(\QQ)`, where if b=0
this is the cusp at infinity.
When present, b must either be Infinity or coercible to an
Integer.
EXAMPLES::
sage: Cusp(2,3)
2/3
sage: Cusp(3,6)
1/2
sage: Cusp(1,0)
Infinity
sage: Cusp(infinity)
Infinity
sage: Cusp(5)
5
sage: Cusp(1/2)
1/2
sage: Cusp(1.5)
3/2
sage: Cusp(int(7))
7
sage: Cusp(1, 2, check=False)
1/2
sage: Cusp('sage', 2.5, check=False) # don't do this!
sage/2.50000000000000
::
sage: I**2
-1
sage: Cusp(I)
Traceback (most recent call last):
...
TypeError: unable to convert I to a cusp
::
sage: a = Cusp(2,3)
sage: loads(a.dumps()) == a
True
::
sage: Cusp(1/3,0)
Infinity
sage: Cusp((1,0))
Infinity
TESTS::
sage: Cusp("1/3", 5)
1/15
sage: Cusp(Cusp(3/5), 7)
3/35
sage: Cusp(5/3, 0)
Infinity
sage: Cusp(3,oo)
0
sage: Cusp((7,3), 5)
7/15
sage: Cusp(int(5), 7)
5/7
::
sage: Cusp(0,0)
Traceback (most recent call last):
...
TypeError: unable to convert (0, 0) to a cusp
::
sage: Cusp(oo,oo)
Traceback (most recent call last):
...
TypeError: unable to convert (+Infinity, +Infinity) to a cusp
::
sage: Cusp(Cusp(oo),oo)
Traceback (most recent call last):
...
TypeError: unable to convert (Infinity, +Infinity) to a cusp
"""
if parent is None:
parent = Cusps
Element.__init__(self, parent)
if not check:
self.__a = a
self.__b = b
return
if b is None:
if isinstance(a, Integer):
self.__a = a
self.__b = ZZ.one()
elif isinstance(a, Rational):
self.__a = a.numer()
self.__b = a.denom()
elif is_InfinityElement(a):
self.__a = ZZ.one()
self.__b = ZZ.zero()
elif isinstance(a, Cusp):
self.__a = a.__a
self.__b = a.__b
elif isinstance(a, integer_types):
self.__a = ZZ(a)
self.__b = ZZ.one()
elif isinstance(a, (tuple, list)):
if len(a) != 2:
raise TypeError("unable to convert %r to a cusp" % a)
if ZZ(a[1]) == 0:
self.__a = ZZ.one()
self.__b = ZZ.zero()
return
try:
r = QQ((a[0], a[1]))
self.__a = r.numer()
self.__b = r.denom()
except (ValueError, TypeError):
raise TypeError("unable to convert %r to a cusp" % a)
else:
try:
r = QQ(a)
self.__a = r.numer()
self.__b = r.denom()
except (ValueError, TypeError):
raise TypeError("unable to convert %r to a cusp" % a)
return
if is_InfinityElement(b):
if is_InfinityElement(a) or (isinstance(a, Cusp) and a.is_infinity()):
raise TypeError("unable to convert (%r, %r) to a cusp" % (a, b))
self.__a = ZZ.zero()
self.__b = ZZ.one()
return
elif not b:
if not a:
raise TypeError("unable to convert (%r, %r) to a cusp" % (a, b))
self.__a = ZZ.one()
self.__b = ZZ.zero()
return
if isinstance(a, Integer) or isinstance(a, Rational):
r = a / ZZ(b)
elif is_InfinityElement(a):
self.__a = ZZ.one()
self.__b = ZZ.zero()
return
elif isinstance(a, Cusp):
if a.__b:
r = a.__a / (a.__b * b)
else:
self.__a = ZZ.one()
self.__b = ZZ.zero()
return
elif isinstance(a, integer_types):
r = ZZ(a) / b
elif isinstance(a, (tuple, list)):
if len(a) != 2:
raise TypeError("unable to convert (%r, %r) to a cusp" % (a, b))
r = ZZ(a[0]) / (ZZ(a[1]) * b)
else:
try:
r = QQ(a) / b
except (ValueError, TypeError):
raise TypeError("unable to convert (%r, %r) to a cusp" % (a, b))
self.__a = r.numer()
self.__b = r.denom()
def __init__(self, a, b=None, parent=None, check=True):
r"""
Create the cusp a/b in `\mathbb{P}^1(\QQ)`, where if b=0
this is the cusp at infinity.
When present, b must either be Infinity or coercible to an
Integer.
EXAMPLES::
sage: Cusp(2,3)
2/3
sage: Cusp(3,6)
1/2
sage: Cusp(1,0)
Infinity
sage: Cusp(infinity)
Infinity
sage: Cusp(5)
5
sage: Cusp(1/2)
1/2
sage: Cusp(1.5)
3/2
sage: Cusp(int(7))
7
sage: Cusp(1, 2, check=False)
1/2
sage: Cusp('sage', 2.5, check=False) # don't do this!
sage/2.50000000000000
::
sage: I**2
-1
sage: Cusp(I)
Traceback (most recent call last):
...
TypeError: Unable to convert I to a Cusp
::
sage: a = Cusp(2,3)
sage: loads(a.dumps()) == a
True
::
sage: Cusp(1/3,0)
Infinity
sage: Cusp((1,0))
Infinity
TESTS::
sage: Cusp("1/3", 5)
1/15
sage: Cusp(Cusp(3/5), 7)
3/35
sage: Cusp(5/3, 0)
Infinity
sage: Cusp(3,oo)
0
sage: Cusp((7,3), 5)
7/15
sage: Cusp(int(5), 7)
5/7
::
sage: Cusp(0,0)
Traceback (most recent call last):
...
TypeError: Unable to convert (0, 0) to a Cusp
::
sage: Cusp(oo,oo)
Traceback (most recent call last):
...
TypeError: Unable to convert (+Infinity, +Infinity) to a Cusp
::
sage: Cusp(Cusp(oo),oo)
Traceback (most recent call last):
...
TypeError: Unable to convert (Infinity, +Infinity) to a Cusp
"""
if parent is None:
parent = Cusps
Element.__init__(self, parent)
if not check:
self.__a = a; self.__b = b
return
if b is None:
if isinstance(a, Integer):
self.__a = a
self.__b = ZZ(1)
elif isinstance(a, Rational):
self.__a = a.numer()
self.__b = a.denom()
elif is_InfinityElement(a):
self.__a = ZZ(1)
self.__b = ZZ(0)
elif isinstance(a, Cusp):
self.__a = a.__a
self.__b = a.__b
elif isinstance(a, (int, long)):
self.__a = ZZ(a)
self.__b = ZZ(1)
elif isinstance(a, (tuple, list)):
if len(a) != 2:
raise TypeError, "Unable to convert %s to a Cusp"%a
if ZZ(a[1]) == 0:
self.__a = ZZ(1)
self.__b = ZZ(0)
return
try:
r = QQ((a[0], a[1]))
self.__a = r.numer()
self.__b = r.denom()
except (ValueError, TypeError):
raise TypeError, "Unable to convert %s to a Cusp"%a
else:
try:
r = QQ(a)
self.__a = r.numer()
self.__b = r.denom()
except (ValueError, TypeError):
raise TypeError, "Unable to convert %s to a Cusp"%a
return
if is_InfinityElement(b):
if is_InfinityElement(a) or (isinstance(a, Cusp) and a.is_infinity()):
raise TypeError, "Unable to convert (%s, %s) to a Cusp"%(a, b)
self.__a = ZZ(0)
self.__b = ZZ(1)
return
elif not b:
if not a:
raise TypeError, "Unable to convert (%s, %s) to a Cusp"%(a, b)
self.__a = ZZ(1)
self.__b = ZZ(0)
return
if isinstance(a, Integer) or isinstance(a, Rational):
r = a / ZZ(b)
elif is_InfinityElement(a):
self.__a = ZZ(1)
self.__b = ZZ(0)
return
elif isinstance(a, Cusp):
if a.__b:
r = a.__a / (a.__b * b)
else:
self.__a = ZZ(1)
self.__b = ZZ(0)
return
elif isinstance(a, (int, long)):
r = ZZ(a) / b
elif isinstance(a, (tuple, list)):
if len(a) != 2:
raise TypeError, "Unable to convert (%s, %s) to a Cusp"%(a, b)
r = ZZ(a[0]) / (ZZ(a[1]) * b)
else:
try:
r = QQ(a) / b
except (ValueError, TypeError):
raise TypeError, "Unable to convert (%s, %s) to a Cusp"%(a, b)
self.__a = r.numer()
self.__b = r.denom()
def __richcmp__(self, other, op):
"""
Implement rich comparison.
We treat two matrix groups as equal if their generators are
the same in the same order. Infinitely-generated groups are
compared by identity.
INPUT:
- ``other`` -- anything
- ``op`` -- comparison operator
OUTPUT:
boolean
EXAMPLES::
sage: G = GL(2,3)
sage: H = MatrixGroup(G.gens())
sage: H == G
True
sage: G == H
True
sage: MS = MatrixSpace(QQ, 2, 2)
sage: G = MatrixGroup([MS(1), MS([1,2,3,4])])
sage: G == G
True
sage: G == MatrixGroup(G.gens())
True
TESTS::
sage: G = groups.matrix.GL(4,2)
sage: H = MatrixGroup(G.gens())
sage: G == H
True
sage: G != H
False
"""
if not is_MatrixGroup(other):
return NotImplemented
if self is other:
return rich_to_bool(op, 0)
lx = self.matrix_space()
rx = other.matrix_space()
if lx != rx:
return richcmp_not_equal(lx, rx, op)
# compare number of generators
try:
n_self = self.ngens()
n_other = other.ngens()
except (AttributeError, NotImplementedError):
return richcmp(id(self), id(other), op)
if n_self != n_other:
return richcmp_not_equal(self, other, op)
from sage.structure.element import is_InfinityElement
if is_InfinityElement(n_self) or is_InfinityElement(n_other):
return richcmp(id(self), id(other), op)
# compact generator matrices
try:
self_gens = self.gens()
other_gens = other.gens()
except (AttributeError, NotImplementedError):
return richcmp(id(self), id(other), op)
assert (n_self == n_other)
for g, h in zip(self_gens, other_gens):
lx = g.matrix()
rx = h.matrix()
if lx != rx:
return richcmp_not_equal(lx, rx, op)
return rich_to_bool(op, 0)
def __richcmp__(self, other, op):
"""
Implement rich comparison.
We treat two matrix groups as equal if their generators are
the same in the same order. Infinitely-generated groups are
compared by identity.
INPUT:
- ``other`` -- anything
- ``op`` -- comparison operator
OUTPUT:
boolean
EXAMPLES::
sage: G = GL(2,3)
sage: H = MatrixGroup(G.gens())
sage: H == G
True
sage: G == H
True
sage: MS = MatrixSpace(QQ, 2, 2)
sage: G = MatrixGroup([MS(1), MS([1,2,3,4])])
sage: G == G
True
sage: G == MatrixGroup(G.gens())
True
TESTS::
sage: G = groups.matrix.GL(4,2)
sage: H = MatrixGroup(G.gens())
sage: G == H
True
sage: G != H
False
"""
if not is_MatrixGroup(other):
return NotImplemented
if self is other:
return rich_to_bool(op, 0)
lx = self.matrix_space()
rx = other.matrix_space()
if lx != rx:
return richcmp_not_equal(lx, rx, op)
# compare number of generators
try:
n_self = self.ngens()
n_other = other.ngens()
except (AttributeError, NotImplementedError):
return richcmp(id(self), id(other), op)
if n_self != n_other:
return richcmp_not_equal(self, other, op)
from sage.structure.element import is_InfinityElement
if is_InfinityElement(n_self) or is_InfinityElement(n_other):
return richcmp(id(self), id(other), op)
# compact generator matrices
try:
self_gens = self.gens()
other_gens = other.gens()
except (AttributeError, NotImplementedError):
return richcmp(id(self), id(other), op)
assert(n_self == n_other)
for g, h in zip(self_gens, other_gens):
lx = g.matrix()
rx = h.matrix()
if lx != rx:
return richcmp_not_equal(lx, rx, op)
return rich_to_bool(op, 0)