def __init__(self, degree, base_ring, category=None):
"""
Base class for matrix groups over generic base rings
You should not use this class directly. Instead, use one of
the more specialized derived classes.
INPUT:
- ``degree`` -- integer. The degree (matrix size) of the
matrix group.
- ``base_ring`` -- ring. The base ring of the matrices.
TESTS::
sage: G = GL(2, QQ)
sage: from sage.groups.matrix_gps.matrix_group import MatrixGroup_generic
sage: isinstance(G, MatrixGroup_generic)
True
"""
assert is_Ring(base_ring)
assert is_Integer(degree)
self._deg = degree
if self._deg <= 0:
raise ValueError('the degree must be at least 1')
if (category is None) and is_FiniteField(base_ring):
from sage.categories.finite_groups import FiniteGroups
category = FiniteGroups()
super(MatrixGroup_generic, self).__init__(base=base_ring, category=category)
def _coerce_(self, x):
"""
EXAMPLES::
>>> s = server(); c = client(s.port)
>>> c._coerce_(False)
0
>>> c._coerce_(True)
1
>>> c._coerce_('lkjdf')
'lkjdf'
>>> c._coerce_(2.5)
2.5
>>> c._coerce_([1,2])
'__pickleeJxrYIot...'
"""
if isinstance(x, bool):
x = int(x)
elif isinstance(x, (str, int, long, float)):
pass
elif is_Integer(x) and x.nbits()<32:
x = int(x)
elif is_RealNumber(x) and x.prec()==53:
return float(x)
elif isinstance(x, unicode):
return str(x)
else:
x = '__pickle' + base64.b64encode(zlib.compress(cPickle.dumps(x, 2)))
return x
def _coerce_(self, x):
"""
EXAMPLES::
>>> s = server(); c = client(s.port)
>>> c._coerce_(False)
0
>>> c._coerce_(True)
1
>>> c._coerce_('lkjdf')
'lkjdf'
>>> c._coerce_(2.5)
2.5
>>> c._coerce_([1,2])
'__pickleeJxrYIot...'
"""
if isinstance(x, bool):
x = int(x)
elif isinstance(x, (str, int, long, float)):
pass
elif x is None:
pass
elif is_Integer(x) and x.nbits() < 32:
x = int(x)
elif is_RealNumber(x) and x.prec() == 53:
return float(x)
elif isinstance(x, unicode):
return str(x)
else:
x = '__pickle' + base64.b64encode(
zlib.compress(cPickle.dumps(x, 2)))
return x
def __init__(self, degree, base_ring, category=None):
"""
Base class for matrix groups over generic base rings
You should not use this class directly. Instead, use one of
the more specialized derived classes.
INPUT:
- ``degree`` -- integer. The degree (matrix size) of the
matrix group.
- ``base_ring`` -- ring. The base ring of the matrices.
TESTS::
sage: G = GL(2, QQ)
sage: from sage.groups.matrix_gps.matrix_group import MatrixGroup_generic
sage: isinstance(G, MatrixGroup_generic)
True
"""
assert is_Ring(base_ring)
assert is_Integer(degree)
self._deg = degree
if self._deg <= 0:
raise ValueError('the degree must be at least 1')
if (category is None) and is_FiniteField(base_ring):
from sage.categories.finite_groups import FiniteGroups
category = FiniteGroups()
super(MatrixGroup_generic, self).__init__(base=base_ring,
category=category)
def __call__(self, P):
r"""
Returns a rational point P in the abstract Homset J(K), given:
0. A point P in J = Jac(C), returning P; 1. A point P on the curve
C such that J = Jac(C), where C is an odd degree model, returning
[P - oo]; 2. A pair of points (P, Q) on the curve C such that J =
Jac(C), returning [P-Q]; 2. A list of polynomials (a,b) such that
`b^2 + h*b - f = 0 mod a`, returning [(a(x),y-b(x))].
EXAMPLES::
sage: P.<x> = PolynomialRing(QQ)
sage: f = x^5 - x + 1; h = x
sage: C = HyperellipticCurve(f,h,'u,v')
sage: P = C(0,1,1)
sage: J = C.jacobian()
sage: Q = J(QQ)(P)
sage: for i in range(6): i*Q
(1)
(u, v - 1)
(u^2, v + u - 1)
(u^2, v + 1)
(u, v + 1)
(1)
::
sage: F.<a> = GF(3)
sage: R.<x> = F[]
sage: f = x^5-1
sage: C = HyperellipticCurve(f)
sage: J = C.jacobian()
sage: X = J(F)
sage: a = x^2-x+1
sage: b = -x +1
sage: c = x-1
sage: d = 0
sage: D1 = X([a,b])
sage: D1
(x^2 + 2*x + 1, y + x + 2)
sage: D2 = X([c,d])
sage: D2
(x + 2, y)
sage: D1+D2
(x^2 + 2*x + 2, y + 2*x + 1)
"""
if isinstance(P, (int, long, Integer)) and P == 0:
R = PolynomialRing(self.value_ring(), 'x')
return JacobianMorphism_divisor_class_field(self, (R(1), R(0)))
elif isinstance(P, (list, tuple)):
if len(P) == 1 and P[0] == 0:
R = PolynomialRing(self.value_ring(), 'x')
return JacobianMorphism_divisor_class_field(self, (R(1), R(0)))
elif len(P) == 2:
P1 = P[0]
P2 = P[1]
if is_Integer(P1) and is_Integer(P2):
R = PolynomialRing(self.value_ring(), 'x')
P1 = R(P1)
P2 = R(P2)
return JacobianMorphism_divisor_class_field(
self, tuple([P1, P2]))
if is_Integer(P1) and is_Polynomial(P2):
R = PolynomialRing(self.value_ring(), 'x')
P1 = R(P1)
return JacobianMorphism_divisor_class_field(
self, tuple([P1, P2]))
if is_Integer(P2) and is_Polynomial(P1):
R = PolynomialRing(self.value_ring(), 'x')
P2 = R(P2)
return JacobianMorphism_divisor_class_field(
self, tuple([P1, P2]))
if is_Polynomial(P1) and is_Polynomial(P2):
return JacobianMorphism_divisor_class_field(self, tuple(P))
if is_SchemeMorphism(P1) and is_SchemeMorphism(P2):
return self(P1) - self(P2)
raise TypeError, "Argument P (= %s) must have length 2." % P
elif isinstance(
P,
JacobianMorphism_divisor_class_field) and self == P.parent():
return P
elif is_SchemeMorphism(P):
x0 = P[0]
y0 = P[1]
R, x = PolynomialRing(self.value_ring(), 'x').objgen()
return self((x - x0, R(y0)))
raise TypeError, "Argument P (= %s) does not determine a divisor class" % P
def __call__(self, P):
r"""
Returns a rational point P in the abstract Homset J(K), given:
0. A point P in J = Jac(C), returning P; 1. A point P on the curve
C such that J = Jac(C), where C is an odd degree model, returning
[P - oo]; 2. A pair of points (P, Q) on the curve C such that J =
Jac(C), returning [P-Q]; 2. A list of polynomials (a,b) such that
`b^2 + h*b - f = 0 mod a`, returning [(a(x),y-b(x))].
EXAMPLES::
sage: P.<x> = PolynomialRing(QQ)
sage: f = x^5 - x + 1; h = x
sage: C = HyperellipticCurve(f,h,'u,v')
sage: P = C(0,1,1)
sage: J = C.jacobian()
sage: Q = J(QQ)(P)
sage: for i in range(6): i*Q
(1)
(u, v - 1)
(u^2, v + u - 1)
(u^2, v + 1)
(u, v + 1)
(1)
::
sage: F.<a> = GF(3)
sage: R.<x> = F[]
sage: f = x^5-1
sage: C = HyperellipticCurve(f)
sage: J = C.jacobian()
sage: X = J(F)
sage: a = x^2-x+1
sage: b = -x +1
sage: c = x-1
sage: d = 0
sage: D1 = X([a,b])
sage: D1
(x^2 + 2*x + 1, y + x + 2)
sage: D2 = X([c,d])
sage: D2
(x + 2, y)
sage: D1+D2
(x^2 + 2*x + 2, y + 2*x + 1)
"""
if isinstance(P, (int, long, Integer)) and P == 0:
R = PolynomialRing(self.value_ring(), "x")
return JacobianMorphism_divisor_class_field(self, (R(1), R(0)))
elif isinstance(P, (list, tuple)):
if len(P) == 1 and P[0] == 0:
R = PolynomialRing(self.value_ring(), "x")
return JacobianMorphism_divisor_class_field(self, (R(1), R(0)))
elif len(P) == 2:
P1 = P[0]
P2 = P[1]
if is_Integer(P1) and is_Integer(P2):
R = PolynomialRing(self.value_ring(), "x")
P1 = R(P1)
P2 = R(P2)
return JacobianMorphism_divisor_class_field(self, tuple([P1, P2]))
if is_Integer(P1) and is_Polynomial(P2):
R = PolynomialRing(self.value_ring(), "x")
P1 = R(P1)
return JacobianMorphism_divisor_class_field(self, tuple([P1, P2]))
if is_Integer(P2) and is_Polynomial(P1):
R = PolynomialRing(self.value_ring(), "x")
P2 = R(P2)
return JacobianMorphism_divisor_class_field(self, tuple([P1, P2]))
if is_Polynomial(P1) and is_Polynomial(P2):
return JacobianMorphism_divisor_class_field(self, tuple(P))
if is_SchemeMorphism(P1) and is_SchemeMorphism(P2):
return self(P1) - self(P2)
raise TypeError, "Argument P (= %s) must have length 2." % P
elif isinstance(P, JacobianMorphism_divisor_class_field) and self == P.parent():
return P
elif is_SchemeMorphism(P):
x0 = P[0]
y0 = P[1]
R, x = PolynomialRing(self.value_ring(), "x").objgen()
return self((x - x0, R(y0)))
raise TypeError, "Argument P (= %s) does not determine a divisor class" % P
