con_field.py

r"""
Projective plane conics over a field

AUTHORS:

- Marco Streng (2010-07-20)

- Nick Alexander (2008-01-08)

"""
#*****************************************************************************
#       Copyright (C) 2008 Nick Alexander <[email protected]>
#       Copyright (C) 2009/2010 Marco Streng <[email protected]>
#
#  Distributed under the terms of the GNU General Public License (GPL)
#
#    This code is distributed in the hope that it will be useful,
#    but WITHOUT ANY WARRANTY; without even the implied warranty of
#    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
#    General Public License for more details.
#
#  The full text of the GPL is available at:
#
#                  http://www.gnu.org/licenses/
#*****************************************************************************

from sage.rings.all import (PolynomialRing, 
                            is_ComplexField, is_RealField,
                            is_Field)
from sage.modules.free_module_element import vector
from sage.structure.sequence import Sequence
from sage.structure.element import is_Vector
from sage.schemes.generic.projective_space import ProjectiveSpace
from sage.matrix.constructor import Matrix
from sage.matrix.matrix import is_Matrix

from sage.schemes.plane_curves.projective_curve import ProjectiveCurve_generic

class ProjectiveConic_field(ProjectiveCurve_generic):
    r"""
    Create a projective plane conic curve over a field.
    See ``Conic`` for full documentation.
    
    EXAMPLES::
    
        sage: K = FractionField(PolynomialRing(QQ, 't'))
        sage: P.<X, Y, Z> = K[]
        sage: Conic(X^2 + Y^2 - Z^2)
        Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t over Rational Field defined by X^2 + Y^2 - Z^2
        
    TESTS::
    
        sage: K = FractionField(PolynomialRing(QQ, 't'))
        sage: Conic([K(1), 1, -1])._test_pickling()
    """
    def __init__(self, A, f):
        r"""
        See ``Conic`` for full documentation.

        EXAMPLES:

        ::

            sage: c = Conic([1, 1, 1]); c
            Projective Conic Curve over Rational Field defined by x^2 + y^2 + z^2
        """
        ProjectiveCurve_generic.__init__(self, A, f)
        self._coefficients = [f[(2,0,0)], f[(1,1,0)], f[(1,0,1)],
                                f[(0,2,0)], f[(0,1,1)], f[(0,0,2)]]
        self._parametrization = None
        self._diagonal_matrix = None
        
        self._rational_point = None




    def _repr_type(self):
        r"""
        Returns ``'Projective Conic'``, which is the first part of the
        plain text representation of this object as output by
        the function ``_repr_`` of the class ``Curve_generic``.

        EXAMPLES::

            sage: c = Conic([1, 1, 1]); c
            Projective Conic Curve over Rational Field defined by x^2 + y^2 + z^2
            sage: c._repr_()
            'Projective Conic Curve over Rational Field defined by x^2 + y^2 + z^2'
            sage: c._repr_type()
            'Projective Conic'
        """
        return "Projective Conic"

    def base_extend(self, S):
        r"""
        Returns the conic over ``S`` given by the same equation as ``self``.

        EXAMPLES::

            sage: c = Conic([1, 1, 1]); c
            Projective Conic Curve over Rational Field defined by x^2 + y^2 + z^2
            sage: c.has_rational_point()
            False
            sage: d = c.base_extend(QuadraticField(-1, 'i')); d
            Projective Conic Curve over Number Field in i with defining polynomial x^2 + 1 defined by x^2 + y^2 + z^2
            sage: d.rational_point(algorithm = 'rnfisnorm')
            (i : 1 : 0)
        """
        if is_Field(S):
            B = self.base_ring()
            if B == S:
                return self
            if not S.has_coerce_map_from(B):
                raise ValueError, "No natural map from the base ring of self " \
                                  "(= %s) to S (= %s)" % (self, S)
            from constructor import Conic
            con = Conic([S(c) for c in self.coefficients()], \
                        self.variable_names())
            if self._rational_point != None:
                pt = [S(c) for c in Sequence(self._rational_point)]
                if not pt == [0,0,0]:
                    # The following line stores the point in the cache
                    # if (and only if) there is no point in the cache.
                    pt = con.point(pt)
            return con
        return ProjectiveCurve_generic.base_extend(self, S)
    
    def cache_point(self, p):
        r"""
        Replace the point in the cache of ``self`` by ``p`` for use
        by ``self.rational_point()`` and ``self.parametrization()``.
        
        EXAMPLES::

            sage: c = Conic([1, -1, 1])
            sage: c.point([15, 17, 8])
            (15/8 : 17/8 : 1)
            sage: c.rational_point()
            (15/8 : 17/8 : 1)
            sage: c.cache_point(c.rational_point(read_cache = False))
            sage: c.rational_point()
            (1 : 1 : 0)
        """
        if isinstance(p, (tuple, list)):
            p = self.point(p)
        self._rational_point = p

    def coefficients(self):
        r"""
        Gives a the `6` coefficients of the conic ``self``
        in lexicographic order.
        
        EXAMPLES::
        
            sage: Conic(QQ, [1,2,3,4,5,6]).coefficients()
            [1, 2, 3, 4, 5, 6]

            sage: P.<x,y,z> = GF(13)[]
            sage: a = Conic(x^2+5*x*y+y^2+z^2).coefficients(); a
            [1, 5, 0, 1, 0, 1]
            sage: Conic(a)
            Projective Conic Curve over Finite Field of size 13 defined by x^2 + 5*x*y + y^2 + z^2
        """
        return self._coefficients
        
        
    def derivative_matrix(self):
        r"""
        Gives the derivative of the defining polynomial of
        the conic ``self``, which is a linear map,
        as a `3 \times 3` matrix.
        
        EXAMPLES:
        
        In characteristic different from `2`, the
        derivative matrix is twice the symmetric matrix:

        ::
        
            sage: c = Conic(QQ, [1,1,1,1,1,0])
            sage: c.symmetric_matrix()
            [  1 1/2 1/2]
            [1/2   1 1/2]
            [1/2 1/2   0]
            sage: c.derivative_matrix()
            [2 1 1]
            [1 2 1]
            [1 1 0]

        An example in characteristic `2`:

        ::

            sage: P.<t> = GF(2)[]
            sage: c = Conic([t, 1, t^2, 1, 1, 0]); c
            Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 2 (using NTL) defined by t*x^2 + x*y + y^2 + t^2*x*z + y*z
            sage: c.is_smooth()
            True
            sage: c.derivative_matrix()
            [  0   1 t^2]
            [  1   0   1]
            [t^2   1   0]
        """
        from sage.matrix.constructor import matrix
        [a,b,c,d,e,f] = self.coefficients()
        return matrix([[ 2*a ,   b ,   c ],
                       [   b , 2*d ,   e ],
                       [   c ,   e , 2*f ]])
       
    def determinant(self):
        r"""
        Returns the determinant of the symmetric matrix that defines
        the conic ``self``.
        
        This is defined only if the base field has characteristic
        different from `2`.

        EXAMPLES:

        ::

            sage: C = Conic([1,2,3,4,5,6])
            sage: C.determinant()
            41/4
            sage: C.symmetric_matrix().determinant()
            41/4

        Determinants are only defined in characteristic different from `2`::

            sage: C = Conic(GF(2), [1, 1, 1, 1, 1, 0])
            sage: C.is_smooth()
            True
            sage: C.determinant()
            Traceback (most recent call last):
            ...
            ValueError: The conic self (= Projective Conic Curve over Finite Field of size 2 defined by x^2 + x*y + y^2 + x*z + y*z) has no symmetric matrix because the base field has characteristic 2
        """
        return self.symmetric_matrix().determinant()

    def diagonal_matrix(self):
        r"""
        Returns a diagonal matrix `D` and a matrix `T` such that `T^t A T = D`
        holds, where `(x, y, z) A (x, y, z)^t` is the defining polynomial
        of the conic ``self``.

        EXAMPLES:

        ::

            sage: c = Conic(QQ, [1,2,3,4,5,6])
            sage: d, t = c.diagonal_matrix(); d, t
            (
            [    1     0     0]  [   1   -1 -7/6]
            [    0     3     0]  [   0    1 -1/3]
            [    0     0 41/12], [   0    0    1]
            )
            sage: t.transpose()*c.symmetric_matrix()*t
            [    1     0     0]
            [    0     3     0]
            [    0     0 41/12]
         
        Diagonal matrices are only defined in characteristic different
        from `2`:
        
        ::

            sage: c = Conic(GF(4, 'a'), [0, 1, 1, 1, 1, 1])
            sage: c.is_smooth()
            True
            sage: c.diagonal_matrix()
            Traceback (most recent call last):
            ...
            ValueError: The conic self (= Projective Conic Curve over Finite Field in a of size 2^2 defined by x*y + y^2 + x*z + y*z + z^2) has no symmetric matrix because the base field has characteristic 2
        """
        A = self.symmetric_matrix()
        B = self.base_ring()
        basis = [vector(B,{2:0,i:1}) for i in range(3)]
        for i in range(3):
            zerovalue = (basis[i]*A*basis[i].column()== 0)
            if zerovalue:
                for j in range(i+1,3):
                    if basis[j]*A*basis[j].column() != 0:
                        b = basis[i]
                        basis[i] = basis[j]
                        basis[j] = b
                        zerovalue = False
            if zerovalue:
                for j in range(i+1,3):
                    if basis[i]*A*basis[j].column() != 0:
                        basis[i] = basis[i]+basis[j]
                        zerovalue = False
            if not zerovalue:
                l = (basis[i]*A*basis[i].column())
                for j in range(i+1,3):
                    basis[j] = basis[j] - \
                               (basis[i]*A*basis[j].column())/l * basis[i]
        T = Matrix(basis).transpose()
        return T.transpose()*A*T, T
        
    def diagonalization(self,names = None):
        r"""
        Returns a diagonal conic `C`, an isomorphism of schemes `M: C` -> ``self``
        and the inverse `N` of `M`.
        
        EXAMPLES::
        
            sage: Conic(GF(5), [1,0,1,1,0,1]).diagonalization()            
            (Projective Conic Curve over Finite Field of size 5 defined by x^2 + y^2 + 2*z^2,
             Scheme morphism:
              From: Projective Conic Curve over Finite Field of size 5 defined by x^2 + y^2 + 2*z^2
              To:   Projective Conic Curve over Finite Field of size 5 defined by x^2 + y^2 + x*z + z^2
              Defn: Defined on coordinates by sending (x : y : z) to
                    (x + 2*z : y : z),
             Scheme morphism:
              From: Projective Conic Curve over Finite Field of size 5 defined by x^2 + y^2 + x*z + z^2
              To:   Projective Conic Curve over Finite Field of size 5 defined by x^2 + y^2 + 2*z^2
              Defn: Defined on coordinates by sending (x : y : z) to
                    (x - 2*z : y : z))

        The diagonalization is only defined in characteristic different
        from 2:
        
        ::

            sage: Conic(GF(2), [1,1,1,1,1,0]).diagonalization()
            Traceback (most recent call last):
            ...
            ValueError: The conic self (= Projective Conic Curve over Finite Field of size 2 defined by x^2 + x*y + y^2 + x*z + y*z) has no symmetric matrix because the base field has characteristic 2
        """
        if names == None:
            names = self.defining_polynomial().parent().variable_names()
        from constructor import Conic
        D, T = self.diagonal_matrix()
        con = Conic(D, names = names)
        return con, con.hom(T, self), self.hom(T.inverse(), con)

    def gens(self):
        r"""
        Returns the generators of the coordinate ring of ``self``.
        
        EXAMPLES:
        
        ::
        
            sage: P.<x,y,z> = QQ[]
            sage: c = Conic(x^2+y^2+z^2)
            sage: c.gens()
            (xbar, ybar, zbar)
            sage: c.defining_polynomial()(c.gens())
            0
            
        The function ``gens()`` is required for the following construction:
        
        ::
        
            sage: C.<a,b,c> = Conic(GF(3), [1, 1, 1])
            sage: C
            Projective Conic Curve over Finite Field of size 3 defined by a^2 + b^2 + c^2

        """
        return self.coordinate_ring().gens()

    def has_rational_point(self, point = False,
                           algorithm = 'default', read_cache = True):
        r"""
        Returns True if and only if the conic ``self``
        has a point over its base field `B`.

        If ``point`` is True, then returns a second output, which is
        a rational point if one exists.
        
        Points are cached whenever they are found. Cached information
        is used if and only if ``read_cache`` is True.
        
        EXAMPLES:

            sage: Conic(RR, [1, 1, 1]).has_rational_point()
            False
            sage: Conic(CC, [1, 1, 1]).has_rational_point()
            True
            
            sage: Conic(RR, [1, 2, -3]).has_rational_point(point = True)
            (True, (1.73205080756888 : 0.000000000000000 : 1.00000000000000))
        """
        if read_cache:
            if self._rational_point is not None:
                if point:
                    return True, self._rational_point
                else:
                    return True
        B = self.base_ring()
        if is_ComplexField(B):
            if point:
                [_,_,_,d,e,f] = self._coefficients
                if d == 0:
                    return True, self.point([0,1,0])
                return True, self.point([0, ((e**2-4*d*f).sqrt()-e)/(2*d), 1],
                                        check = False)
            return True
        if is_RealField(B):
            D, T = self.diagonal_matrix()
            [a, b, c] = [D[0,0], D[1,1], D[2,2]]
            if a == 0:
                ret = True, self.point(T*vector([1,0,0]), check = False)
            elif a*c <= 0:
                ret = True, self.point(T*vector([(-c/a).sqrt(),0,1]),
                                       check = False)
            elif b == 0:
                ret = True, self.point(T*vector([0,1,0]), check = False)
            elif b*c <= 0:
                ret = True, self.point(T*vector([0,(-c/b).sqrt(),0,1]),
                                       check = False)
            else:
                ret = False, None
            if point:
                return ret
            return ret[0]
        raise NotImplementedError, "has_rational_point not implemented for " \
                                   "conics over base field %s" % B 
        
    def has_singular_point(self, point = False):
        r"""
        Return True if and only if the conic ``self`` has a rational
        singular point.
        
        If ``point`` is True, then also return a rational singular
        point (or ``None`` if no such point exists).
        
        EXAMPLES:
        
        ::
            
            sage: c = Conic(QQ, [1,0,1]); c
            Projective Conic Curve over Rational Field defined by x^2 + z^2
            sage: c.has_singular_point(point = True)
            (True, (0 : 1 : 0))
            
            sage: P.<x,y,z> = GF(7)[]
            sage: e = Conic((x+y+z)*(x-y+2*z)); e
            Projective Conic Curve over Finite Field of size 7 defined by x^2 - y^2 + 3*x*z + y*z + 2*z^2
            sage: e.has_singular_point(point = True)
            (True, (2 : 4 : 1))

            sage: Conic([1, 1, -1]).has_singular_point()
            False
            sage: Conic([1, 1, -1]).has_singular_point(point = True)
            (False, None)

        ``has_singular_point`` is not implemented over all fields
        of characteristic `2`. It is implemented over finite fields.

        ::

            sage: F.<a> = FiniteField(8)
            sage: Conic([a, a+1, 1]).has_singular_point(point = True)
            (True, (a + 1 : 0 : 1))

            sage: P.<t> = GF(2)[]
            sage: C = Conic(P, [t,t,1]); C
            Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 2 (using NTL) defined by t*x^2 + t*y^2 + z^2
            sage: C.has_singular_point(point = False)
            Traceback (most recent call last):
            ...
            NotImplementedError: Sorry, find singular point on conics not implemented over all fields of characteristic 2.
        """
        if not point:
           ret = self.has_singular_point(point = True)
           return ret[0]
        B = self.base_ring()
        if B.characteristic() == 2:
            [a,b,c,d,e,f] = self.coefficients()
            if b == 0 and c == 0 and e == 0:
                for i in range(3):
                    if [a, d, f][i] == 0:
                        return True, self.point(vector(B, {2:0, i:1}))
                if hasattr(a/f, 'is_square') and hasattr(a/f, 'sqrt'):
                    if (a/f).is_square():
                        return True, self.point([1,0,(a/f).sqrt()])
                    if (d/f).is_square():
                        return True, self.point([0,1,(d/f).sqrt()])
                raise NotImplementedError, "Sorry, find singular point on conics not implemented over all fields of characteristic 2."
            pt = [e, c, b]
            if self.defining_polynomial()(pt) == 0:
                return True, self.point(pt)
            return False, None
        D = self.symmetric_matrix()
        if D.determinant() == 0:
            return True, self.point(Sequence(D.right_kernel().gen()))
        return False, None

    def hom(self, x, Y=None):
        r"""
        Return the scheme morphism from ``self`` to ``Y`` defined by ``x``.
        Here ``x`` can be a matrix or a sequence of polynomials.
        If ``Y`` is omitted, then a natural image is found if possible.

        EXAMPLES:
        
        Here are a few Morphisms given by matrices. In the first
        example, ``Y`` is omitted, in the second example, ``Y`` is specified.

        ::
            
            sage: c = Conic([-1, 1, 1])
            sage: h = c.hom(Matrix([[1,1,0],[0,1,0],[0,0,1]])); h
            Scheme morphism:
              From: Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2
              To:   Projective Conic Curve over Rational Field defined by -x^2 + 2*x*y + z^2
              Defn: Defined on coordinates by sending (x : y : z) to
                    (x + y : y : z)
            sage: h([-1, 1, 0])
            (0 : 1 : 0)

            sage: c = Conic([-1, 1, 1])
            sage: d = Conic([4, 1, -1])
            sage: c.hom(Matrix([[0, 0, 1/2], [0, 1, 0], [1, 0, 0]]), d)
            Scheme morphism:
              From: Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2
              To:   Projective Conic Curve over Rational Field defined by 4*x^2 + y^2 - z^2
              Defn: Defined on coordinates by sending (x : y : z) to
                    (1/2*z : y : x)

        ``ValueError`` is raised if the wrong codomain ``Y`` is specified:

        ::
        
            sage: c = Conic([-1, 1, 1])
            sage: c.hom(Matrix([[0, 0, 1/2], [0, 1, 0], [1, 0, 0]]), c)
            Traceback (most recent call last):
            ...
            ValueError: The matrix x (= [  0   0 1/2]
            [  0   1   0]
            [  1   0   0]) does not define a map from self (= Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2) to Y (= Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2)
        """
        if is_Matrix(x):    
            from constructor import Conic
            y = x.inverse()
            A = y.transpose()*self.matrix()*y
            im = Conic(A)
            if Y == None:
                Y = im
            else:
                q = Y.defining_polynomial()/im.defining_polynomial()
                if not (q.numerator().is_constant()
                        and q.denominator().is_constant()):
                    raise ValueError, "The matrix x (= %s) does not define a " \
                                      "map from self (= %s) to Y (= %s)" % \
                                      (x, self, Y)
            x = Sequence(x*vector(self.ambient_space().gens()))
            return self.Hom(Y)(x, check = False)
        return ProjectiveCurve_generic.hom(self, x, Y)            
            
       
    def is_diagonal(self):
        r"""
        Return True if and only if the conic has the form
        `a*x^2 + b*y^2 + c*z^2`.

        EXAMPLES:

        ::

            sage: c=Conic([1,1,0,1,0,1]); c
            Projective Conic Curve over Rational Field defined by x^2 + x*y + y^2 + z^2
            sage: d,t = c.diagonal_matrix()
            sage: c.is_diagonal()
            False
            sage: c.diagonalization()[0].is_diagonal()
            True
        """
        return all([self.coefficients()[i] == 0 for i in [1,2,4]])


    def is_smooth(self):
        r"""
        Returns True if and only if ``self`` is smooth.

        EXAMPLES:

        ::

            sage: Conic([1,-1,0]).is_smooth()
            False
            sage: Conic(GF(2),[1,1,1,1,1,0]).is_smooth()
            True
        """
        if self.base_ring().characteristic() == 2:
            [a,b,c,d,e,f] = self.coefficients()
            if b == 0 and c == 0 and e == 0:
                return False
            return self.defining_polynomial()([e, c, b]) != 0
        return self.determinant() != 0


    def _magma_init_(self, magma):
        """
        Internal function. Returns a string to initialize this
        conic in the Magma subsystem.
        
        EXAMPLES::
        
            sage: C = Conic(QQ, [1,2,3])
            sage: C._magma_init_(magma)                          # optional - magma
            'Conic([_sage_ref1|1/1,2/1,3/1,0/1,0/1,0/1])'
            sage: C = Conic(GF(41), [-1,2,5])                    # optional - magma
            sage: C._magma_init_(magma)                          # optional - magma
            'Conic([_sage_ref2|GF(41)!40,GF(41)!2,GF(41)!5,GF(41)!0,GF(41)!0,GF(41)!0])'
            sage: F.<a> = GF(25)
            sage: C = Conic([3,0,1,4,a,2])
            sage: C
            Projective Conic Curve over Finite Field in a of size 5^2 defined by -2*x^2 - y^2 + x*z + (a)*y*z + 2*z^2
            sage: magma(C)                                       # optional - magma
            Conic over GF(5^2) defined by
            3*X^2 + 4*Y^2 + X*Z + a*Y*Z + 2*Z^2
            sage: magma(Conic([1/2,2/3,-4/5,6/7,8/9,-10/11]))    # optional - magma
            Conic over Rational Field defined by
            1/2*X^2 + 2/3*X*Y + 6/7*Y^2 - 4/5*X*Z + 8/9*Y*Z - 10/11*Z^2
            sage: R.<x> = Frac(QQ['x'])
            sage: magma(Conic([x,1+x,1-x]))                      # optional - magma
            Conic over Univariate rational function field over Rational Field defined by
            x*X^2 + (x + 1)*Y^2 + (-x + 1)*Z^2
            sage: P.<x> = QQ[]
            sage: K.<b> = NumberField(x^3+x+1)
            sage: magma(Conic([b,1,2]))                          # optional - magma
            Conic over Number Field with defining polynomial x^3 + x + 1 over the Rational Field defined by
            b*X^2 + Y^2 + 2*Z^2
        """
        kmn = magma(self.base_ring())._ref()
        coeffs = self.coefficients()
        magma_coeffs = [coeffs[i]._magma_init_(magma) for i in [0, 3, 5, 1, 4, 2]]
        return 'Conic([%s|%s])' % (kmn,','.join(magma_coeffs))


    def matrix(self):
        r"""
        Returns a matrix `M` such that `(x, y, z) M (x, y, z)^t`
        is the defining equation of ``self``.

        The matrix `M` is upper triangular if the base field has
        characteristic `2` and symmetric otherwise.

        EXAMPLES::
        
            sage: R.<x, y, z> = QQ[]
            sage: C = Conic(x^2 + x*y + y^2 + z^2)
            sage: C.matrix()
            [  1 1/2   0]
            [1/2   1   0]
            [  0   0   1]

            sage: R.<x, y, z> = GF(2)[]
            sage: C = Conic(x^2 + x*y + y^2 + x*z + z^2)
            sage: C.matrix()
            [1 1 1]
            [0 1 0]
            [0 0 1]
        """
        if self.base_ring().characteristic() == 2:
            return self.upper_triangular_matrix()
        return self.symmetric_matrix()
        
    _matrix_ = matrix

    def parametrization(self, point=None, morphism=True):
        r"""
        Return a parametrization `f` of ``self`` together with the
        inverse of `f`.

        If ``point`` is specified, then that point is used
        for the parametrization. Otherwise, use ``self.rational_point()``
        to find a point.
            
        If ``morphism`` is True, then `f` is returned in the form
        of a Scheme morphism. Otherwise, it is a tuple of polynomials
        that gives the parametrization.
        
        EXAMPLES:

        An example over a finite field ::
        
            sage: c = Conic(GF(2), [1,1,1,1,1,0])
            sage: c.parametrization()
            (Scheme morphism:
              From: Projective Space of dimension 1 over Finite Field of size 2
              To:   Projective Conic Curve over Finite Field of size 2 defined by x^2 + x*y 
            + y^2 + x*z + y*z
              Defn: Defined on coordinates by sending (x : y) to
                    (x*y + y^2 : x^2 + x*y : x^2 + x*y + y^2),
             Scheme morphism:
              From: Projective Conic Curve over Finite Field of size 2 defined by x^2 + x*y 
            + y^2 + x*z + y*z
              To:   Projective Space of dimension 1 over Finite Field of size 2
              Defn: Defined on coordinates by sending (x : y : z) to
                    (y : x))

        An example with ``morphism = False`` ::

            sage: R.<x,y,z> = QQ[]
            sage: C = Curve(7*x^2 + 2*y*z + z^2)
            sage: (p, i) = C.parametrization(morphism = False); (p, i)
            ([-2*x*y, 7*x^2 + y^2, -2*y^2], [-1/2*x, -1/2*z])
            sage: C.defining_polynomial()(p)
            0
            sage: i[0](p) / i[1](p)
            x/y
            
        A ``ValueError`` is raised if ``self`` has no rational point ::

            sage: C = Conic(x^2 + y^2 + 7*z^2)
            sage: C.parametrization()
            Traceback (most recent call last):
            ...
            ValueError: Conic Projective Conic Curve over Rational Field defined by x^2 + y^2 + 7*z^2 has no rational points over Rational Field!
            
        A ``ValueError`` is raised if ``self`` is not smooth ::
        
            sage: C = Conic(x^2 + y^2)
            sage: C.parametrization()
            Traceback (most recent call last):
            ...
            ValueError: The conic self (=Projective Conic Curve over Rational Field defined by x^2 + y^2) is not smooth, hence does not have a parametrization.
        """
        if (not self._parametrization is None) and not point:
            par = self._parametrization
        else:
            if not self.is_smooth():
                raise ValueError, "The conic self (=%s) is not smooth, hence does not have a parametrization." % self
            if point == None:
                point = self.rational_point()
            point = Sequence(point)
            B = self.base_ring()
            Q = PolynomialRing(B, 'x,y')
            [x, y] = Q.gens()
            gens = self.ambient_space().gens()
            P = PolynomialRing(B, 4, ['X', 'Y', 'T0', 'T1'])
            [X, Y, T0, T1] = P.gens()
            c3 = [j for j in range(2,-1,-1) if point[j] != 0][0]
            c1 = [j for j in range(3) if j != c3][0]
            c2 = [j for j in range(3) if j != c3 and j != c1][0]
            L = [0,0,0]
            L[c1] = Y*T1*point[c1] + Y*T0
            L[c2] = Y*T1*point[c2] + X*T0
            L[c3] = Y*T1*point[c3]
            bezout = P(self.defining_polynomial()(L) / T0)
            t = [bezout([x,y,0,-1]),bezout([x,y,1,0])]
            par = (tuple([Q(p([x,y,t[0],t[1]])/y) for  p in L]),
                   tuple([gens[m]*point[c3]-gens[c3]*point[m]
                       for m in [c2,c1]]))
            if self._parametrization is None:
                self._parametrization = par
        if not morphism:
            return par
        P1 = ProjectiveSpace(self.base_ring(), 1, 'x,y')
        return P1.hom(par[0],self), self.Hom(P1)(par[1], check = False)
        
    def point(self, v, check=True):
        r"""
        Constructs a point on ``self`` corresponding to the input ``v``.
        
        If ``check`` is True, then checks if ``v`` defines a valid
        point on ``self``.
        
        If no rational point on ``self`` is known yet, then also caches the point
        for use by ``self.rational_point()`` and ``self.parametrization()``.

        EXAMPLES ::

            sage: c = Conic([1, -1, 1])
            sage: c.point([15, 17, 8])
            (15/8 : 17/8 : 1)
            sage: c.rational_point()
            (15/8 : 17/8 : 1)
            sage: d = Conic([1, -1, 1])
            sage: d.rational_point()
            (1 : 1 : 0)
        """
        if is_Vector(v):
            v = Sequence(v)
        p = ProjectiveCurve_generic.point(self, v, check=check)
        if self._rational_point is None:
            self._rational_point = p
        return p

        
    def random_rational_point(self, *args1, **args2):
        r"""
        Return a random rational point of the conic ``self``.
        
        ALGORITHM:
        
            1. Compute a parametrization `f` of ``self`` using
               ``self.parametrization()``.
            2. Computes a random point `(x:y)` on the projective
               line.
            3. Output `f(x:y)`.
        
        The coordinates x and y are computed using
        ``B.random_element``, where ``B`` is the base field of
        ``self`` and additional arguments to ``random_rational_point``
        are passed to ``random_element``.
            
        If the base field is a finite field, then the
        output is uniformly distributed over the points of self.

        EXAMPLES ::
            
            sage: c = Conic(GF(2), [1,1,1,1,1,0])
            sage: [c.random_rational_point() for i in range(10)] # output is random
            [(1 : 0 : 1), (1 : 0 : 1), (1 : 0 : 1), (0 : 1 : 1), (1 : 0 : 1), (0 : 0 : 1), (1 : 0 : 1), (1 : 0 : 1), (0 : 0 : 1), (1 : 0 : 1)]
            
            sage: d = Conic(QQ, [1, 1, -1])
            sage: d.random_rational_point(den_bound = 1, num_bound = 5) # output is random
            (-24/25 : 7/25 : 1)
            
            sage: Conic(QQ, [1, 1, 1]).random_rational_point()
            Traceback (most recent call last):
            ...
            ValueError: Conic Projective Conic Curve over Rational Field defined by x^2 + y^2 + z^2 has no rational points over Rational Field!

        """
        if not self.is_smooth():
            raise NotImplementedError, "Sorry, random points not implemented " \
                                       "for non-smooth conics"
        par = self.parametrization()
        x = 0
        y = 0
        B = self.base_ring()
        while x == 0 and y == 0:
            x = B.random_element(*args1, **args2)
            y = B.random_element(*args1, **args2)
        return par[0]([x,y])


    def rational_point(self, algorithm = 'default', read_cache = True):
        r"""
        Return a point on ``self`` defined over the base field.

        Raises ``ValueError`` if no rational point exists.

        See ``self.has_rational_point`` for the algorithm used
        and for the use of the parameters ``algorithm`` and ``read_cache``.
         
        EXAMPLES:

        Examples over `\QQ` ::
        
            sage: R.<x,y,z> = QQ[]
            sage: C = Conic(7*x^2 + 2*y*z + z^2)
            sage: C.rational_point()
            (0 : 1 : 0)

            sage: C = Conic(x^2 + 2*y^2 + z^2)
            sage: C.rational_point()
            Traceback (most recent call last):
            ...
            ValueError: Conic Projective Conic Curve over Rational Field defined by x^2 + 2*y^2 + z^2 has no rational points over Rational Field!

            sage: C = Conic(x^2 + y^2 + 7*z^2)
            sage: C.rational_point(algorithm = 'rnfisnorm')
            Traceback (most recent call last):
            ...
            ValueError: Conic Projective Conic Curve over Rational Field defined by x^2 + y^2 + 7*z^2 has no rational points over Rational Field!

        Examples over number fields ::
            
            sage: P.<x> = QQ[]
            sage: L.<b> = NumberField(x^3-5)
            sage: C = Conic(L, [3, 2, -5])
            sage: C.rational_point(algorithm = 'rnfisnorm')  # long time (1/2 second), output is random
            (37632*b^2 + 88168*b - 260801 : -99528*b^2 + 153348*b + 28799 : 1)
        
        Examples over finite fields ::
        
            sage: F.<a> = FiniteField(7^20)
            sage: C = Conic([1, a, -5]); C
            Projective Conic Curve over Finite Field in a of size 7^20 defined by x^2 + (a)*y^2 + 2*z^2
            sage: C.rational_point()  # output is random
            (4*a^19 + 5*a^18 + 4*a^17 + a^16 + 6*a^15 + 3*a^13 + 6*a^11 + a^9 + 3*a^8 + 2*a^7 + 4*a^6 + 3*a^5 + 3*a^4 + a^3 + a + 6 : 5*a^18 + a^17 + a^16 + 6*a^15 + 4*a^14 + a^13 + 5*a^12 + 5*a^10 + 2*a^9 + 6*a^8 + 6*a^7 + 6*a^6 + 2*a^4 + 3 : 1)
             
        Examples over `\RR` and `\CC` ::

            sage: Conic(CC, [1, 2, 3]).rational_point()
            (0 : 1.22474487139159*I : 1)

            sage: Conic(RR, [1, 1, 1]).rational_point()
            Traceback (most recent call last):
            ...
            ValueError: Conic Projective Conic Curve over Real Field with 53 bits of precision defined by x^2 + y^2 + z^2 has no rational points over Real Field with 53 bits of precision!
        """
        bl,pt = self.has_rational_point(point = True, algorithm = algorithm,
                                        read_cache = read_cache)
        if bl:
            return pt
        raise ValueError, "Conic %s has no rational points over %s!" % \
                          (self, self.ambient_space().base_ring())


    def singular_point(self):
        r"""
        Returns a singular rational point of ``self``
        
        EXAMPLES:
        
        ::

            sage: Conic(GF(2), [1,1,1,1,1,1]).singular_point()
            (1 : 1 : 1)

        ``ValueError`` is raised if the conic has no rational singular point
            
        ::

            sage: Conic(QQ, [1,1,1,1,1,1]).singular_point()
            Traceback (most recent call last):
            ...
            ValueError: The conic self (= Projective Conic Curve over Rational Field defined by x^2 + x*y + y^2 + x*z + y*z + z^2) has no rational singular point
        """
        b = self.has_singular_point(point = True)
        if not b[0]:
            raise ValueError, "The conic self (= %s) has no rational " \
                              "singular point" % self
        return b[1]

    def symmetric_matrix(self):
        r"""
        The symmetric matrix `M` such that `(x y z) M (x y z)^t`
        is the defining equation of ``self``.
        
        EXAMPLES ::
        
            sage: R.<x, y, z> = QQ[]
            sage: C = Conic(x^2 + x*y/2 + y^2 + z^2)
            sage: C.symmetric_matrix()
            [  1 1/4   0]
            [1/4   1   0]
            [  0   0   1]

            sage: C = Conic(x^2 + 2*x*y + y^2 + 3*x*z + z^2)
            sage: v = vector([x, y, z])
            sage: v * C.symmetric_matrix() * v
            x^2 + 2*x*y + y^2 + 3*x*z + z^2
        """
        [a,b,c,d,e,f] = self.coefficients()
        if self.base_ring().characteristic() == 2:
            if b == 0 and c == 0 and e == 0:
                return matrix([[a,0,0],[0,d,0],[0,0,f]])
            raise ValueError, "The conic self (= %s) has no symmetric matrix " \
                              "because the base field has characteristic 2" % \
                              self
        from sage.matrix.constructor import matrix
        return matrix([[  a , b/2, c/2 ],
                       [ b/2,  d , e/2 ],
                       [ c/2, e/2,  f  ]])


    def upper_triangular_matrix(self):
        r"""
        The upper-triangular matrix `M` such that `(x y z) M (x y z)^t`
        is the defining equation of ``self``.

        EXAMPLES::

            sage: R.<x, y, z> = QQ[]
            sage: C = Conic(x^2 + x*y + y^2 + z^2)
            sage: C.upper_triangular_matrix()
            [1 1 0]
            [0 1 0]
            [0 0 1]

            sage: C = Conic(x^2 + 2*x*y + y^2 + 3*x*z + z^2)
            sage: v = vector([x, y, z])
            sage: v * C.upper_triangular_matrix() * v
            x^2 + 2*x*y + y^2 + 3*x*z + z^2
        """
        from sage.matrix.constructor import matrix
        [a,b,c,d,e,f] = self.coefficients()
        return matrix([[ a, b, c ],
                       [ 0, d, e ],
                       [ 0, 0, f ]])
    
    def variable_names(self):
        r"""
        Returns the variable names of the defining polynomial
        of ``self``.

        EXAMPLES:

        ::

            sage: c=Conic([1,1,0,1,0,1], 'x,y,z')
            sage: c.variable_names()
            ('x', 'y', 'z')
            sage: c.variable_name()
            'x'
            
        The function ``variable_names()`` is required
        for the following construction:
        
        ::
        
            sage: C.<p,q,r> = Conic(QQ, [1, 1, 1])
            sage: C
            Projective Conic Curve over Rational Field defined by p^2 + q^2 + r^2

        """
        return self.defining_polynomial().parent().variable_names()