def create_key_and_extra_args(self, X, Y, category=None, base=ZZ,
                                  check=True):
        """
        Create a key that uniquely determines the Hom-set.
        
        INPUT:
        
        - ``X`` -- a scheme. The domain of the morphisms.
        
        - ``Y`` -- a scheme. The codomain of the morphisms.
        
        - ``category`` -- a category for the Hom-sets (default: schemes over
          given base).

        - ``base`` -- a scheme or a ring. The base scheme of domain
          and codomain schemes. If a ring is specified, the spectrum
          of that ring will be used as base scheme.

        - ``check`` -- boolean (default: ``True``).

        EXAMPLES::
        
            sage: A2 = AffineSpace(QQ,2)
            sage: A3 = AffineSpace(QQ,3)
            sage: A3.Hom(A2)    # indirect doctest
            Set of morphisms
              From: Affine Space of dimension 3 over Rational Field
              To:   Affine Space of dimension 2 over Rational Field
            sage: from sage.schemes.generic.homset import SchemeHomsetFactory
            sage: SHOMfactory = SchemeHomsetFactory('test')
            sage: key, extra = SHOMfactory.create_key_and_extra_args(A3,A2,check=False)
            sage: key
            (..., ..., Category of schemes over Integer Ring)
            sage: extra
            {'Y': Affine Space of dimension 2 over Rational Field,
             'X': Affine Space of dimension 3 over Rational Field,
             'base_ring': Integer Ring, 'check': False}
        """
        if not is_Scheme(X) and is_CommutativeRing(X): 
            X = Spec(X)
        if not is_Scheme(Y) and is_CommutativeRing(Y): 
            Y = Spec(Y)
        if is_Spec(base):
            base_spec = base
            base_ring = base.coordinate_ring()
        elif is_CommutativeRing(base): 
            base_spec = Spec(base)
            base_ring = base
        else:
            raise ValueError(
                        'The base must be a commutative ring or its spectrum.')
        if not category:
            from sage.categories.schemes import Schemes
            category = Schemes(base_spec)
        key = tuple([id(X), id(Y), category])
        extra = {'X':X, 'Y':Y, 'base_ring':base_ring, 'check':check}
        return key, extra

    def __init__(self, X, coordinates, check=True):
        r"""
        See :class:`SchemeMorphism_point_toric_field` for documentation.

        TESTS::

            sage: fan = FaceFan(lattice_polytope.octahedron(2))
            sage: P1xP1 = ToricVariety(fan)
            sage: P1xP1(1,2,3,4)
            [1 : 2 : 3 : 4]
        """
        # Convert scheme to its set of points over the base ring
        if is_Scheme(X):
            X = X(X.base_ring())
        super(SchemeMorphism_point_toric_field, self).__init__(X)
        if check:
            # Verify that there are the right number of coords
            # Why is it not done in the parent?
            if is_SchemeMorphism(coordinates):
                coordinates = list(coordinates)
            if not isinstance(coordinates, (list, tuple)):
                raise TypeError("coordinates must be a scheme point, list, "
                                "or tuple. Got %s" % coordinates)
            d = X.codomain().ambient_space().ngens()
            if len(coordinates) != d:
                raise ValueError("there must be %d coordinates! Got only %d: "
                                 "%s" % (d, len(coordinates), coordinates))
            # Make sure the coordinates all lie in the appropriate ring
            coordinates = Sequence(coordinates, X.value_ring())
            # Verify that the point satisfies the equations of X.
            X.codomain()._check_satisfies_equations(coordinates)
        self._coords = coordinates

def Jacobian(X, **kwds):
    """
    Return the Jacobian.

    INPUT:

    - ``X`` -- polynomial, algebraic variety, or anything else that
      has a Jacobian elliptic curve.

    - ``kwds`` -- optional keyword arguments.

    The input ``X`` can be one of the following:

    * A polynomial, see :func:`Jacobian_of_equation` for details.

    * A curve, see :func:`Jacobian_of_curve` for details.

    EXAMPLES::

        sage: R.<u,v,w> = QQ[]
        sage: Jacobian(u^3+v^3+w^3)
        Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field

        sage: C = Curve(u^3+v^3+w^3)
        sage: Jacobian(C)
        Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field

        sage: P2.<u,v,w> = ProjectiveSpace(2, QQ)
        sage: C = P2.subscheme(u^3+v^3+w^3)
        sage: Jacobian(C)
        Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field

        sage: Jacobian(C, morphism=True)
        Scheme morphism:
          From: Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
          u^3 + v^3 + w^3
          To:   Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field
          Defn: Defined on coordinates by sending (u : v : w) to
                (-u^4*v^4*w - u^4*v*w^4 - u*v^4*w^4 :
                1/2*u^6*v^3 - 1/2*u^3*v^6 - 1/2*u^6*w^3 + 1/2*v^6*w^3 + 1/2*u^3*w^6 - 1/2*v^3*w^6 :
                u^3*v^3*w^3)
    """
    try:
        return X.jacobian(**kwds)
    except AttributeError:
        pass

    morphism = kwds.pop('morphism', False)
    from sage.rings.polynomial.multi_polynomial_element import is_MPolynomial
    if is_MPolynomial(X):
        if morphism:
            from sage.schemes.curves.constructor import Curve
            return Jacobian_of_equation(X, curve=Curve(X), **kwds)
        else:
            return Jacobian_of_equation(X, **kwds)

    from sage.schemes.generic.scheme import is_Scheme
    if is_Scheme(X) and X.dimension() == 1:
        return Jacobian_of_curve(X, morphism=morphism, **kwds)

def enum_affine_number_field(X, B):
    """
    Enumerates affine points on scheme ``X`` defined over a number field. Simply checks all of the
    points of absolute height up to ``B`` and adds those that are on the scheme to the list.

    INPUT:

    - ``X`` - a scheme defined over a number field.

    - ``B`` - a real number.

    OUTPUT:

     - a list containing the affine points of ``X`` of absolute height up to ``B``,
       sorted.

    EXAMPLES::

        sage: from sage.schemes.affine.affine_rational_point import enum_affine_number_field
        sage: u = QQ['u'].0
        sage: K = NumberField(u^2 + 2, 'v')
        sage: A.<x,y,z> = AffineSpace(K, 3)
        sage: X = A.subscheme([y^2 - x])
        sage: enum_affine_number_field(X(K), 4)
        [(0, 0, -1), (0, 0, -v), (0, 0, -1/2*v), (0, 0, 0), (0, 0, 1/2*v), (0, 0, v), (0, 0, 1),
        (1, -1, -1), (1, -1, -v), (1, -1, -1/2*v), (1, -1, 0), (1, -1, 1/2*v), (1, -1, v), (1, -1, 1),
        (1, 1, -1), (1, 1, -v), (1, 1, -1/2*v), (1, 1, 0), (1, 1, 1/2*v), (1, 1, v), (1, 1, 1)]

    ::

        sage: u = QQ['u'].0
        sage: K = NumberField(u^2 + 3, 'v')
        sage: A.<x,y> = AffineSpace(K, 2)
        sage: X=A.subscheme(x-y)
        sage: from sage.schemes.affine.affine_rational_point import enum_affine_number_field
        sage: enum_affine_number_field(X, 3)
        [(-1, -1), (-1/2*v - 1/2, -1/2*v - 1/2), (1/2*v - 1/2, 1/2*v - 1/2), (0, 0), (-1/2*v + 1/2, -1/2*v + 1/2),
        (1/2*v + 1/2, 1/2*v + 1/2), (1, 1)]
    """
    from sage.schemes.affine.affine_space import is_AffineSpace
    if(is_Scheme(X)):
        if (not is_AffineSpace(X.ambient_space())):
            raise TypeError("ambient space must be affine space over a number field")
        X = X(X.base_ring())
    else:
        if (not is_AffineSpace(X.codomain().ambient_space())):
            raise TypeError("codomain must be affine space over a number field")

    R = X.codomain().ambient_space()

    pts = []
    for P in R.points_of_bounded_height(B):
        try:
            pts.append(X(P))
        except TypeError:
            pass
    pts.sort()
    return pts

def enum_projective_number_field(X,B):
    """
    Enumerates projective points on scheme ``X`` defined over a number field. Simply checks all of the
    points of absolute height of at most ``B`` and adds those that are on the scheme to the list.

    INPUT:

    - ``X`` - a scheme defined over a number field

    - ``B`` - a real number

    OUTPUT:

     - a list containing the projective points of ``X`` of absolute height up to ``B``,
       sorted.

    EXAMPLES::

        sage: from sage.schemes.projective.projective_rational_point import enum_projective_number_field
        sage: u = QQ['u'].0
        sage: K = NumberField(u^3 - 5,'v')
        sage: P.<x,y,z> = ProjectiveSpace(K, 2)
        sage: X = P.subscheme([x - y])
        sage: enum_projective_number_field(X(K),125)
        [(0 : 0 : 1), (-1 : -1 : 1), (1 : 1 : 1), (-1/5*v^2 : -1/5*v^2 : 1), (-v : -v : 1),
        (1/5*v^2 : 1/5*v^2 : 1), (v : v : 1), (1 : 1 : 0)]

    ::

        sage: u = QQ['u'].0
        sage: K = NumberField(u^2 + 3,'v')
        sage: A.<x,y> = ProjectiveSpace(K,1)
        sage: X=A.subscheme(x-y)
        sage: from sage.schemes.projective.projective_rational_point import enum_projective_number_field
        sage: enum_projective_number_field(X,3)
        [(1 : 1)]
    """
    from sage.schemes.projective.projective_space import is_ProjectiveSpace
    if(is_Scheme(X)):
        if (not is_ProjectiveSpace(X.ambient_space())):
            raise TypeError("Ambient space must be projective space over a number field")
        X = X(X.base_ring())
    else:
        if (not is_ProjectiveSpace(X.codomain().ambient_space())):
            raise TypeError("Codomain must be projective space over a number field")

    R = X.codomain().ambient_space()

    pts = []

    for P in R.points_of_bounded_height(B):
        try:
            pts.append(X(P))
        except TypeError:
            pass
    pts.sort()
    return pts

    def __call__(self, x):
        """
        Call syntax for Spec.

        INPUT/OUTPUT:

        The argument ``x`` must be one of the following:

        - a prime ideal of the coordinate ring; the output will
          be the corresponding point of X

        - an element (or list of elements) of the coordinate ring
          which generates a prime ideal; the output will be the
          corresponding point of X

        - a ring or a scheme S; the output will be the set X(S) of
          S-valued points on X

        EXAMPLES::

            sage: S = Spec(ZZ)
            sage: P = S(3); P
            Point on Spectrum of Integer Ring defined by the Principal ideal (3) of Integer Ring
            sage: type(P)
            <class 'sage.schemes.generic.point.SchemeTopologicalPoint_prime_ideal'>
            sage: S(ZZ.ideal(next_prime(1000000)))
            Point on Spectrum of Integer Ring defined by the Principal ideal (1000003) of Integer Ring

            sage: R.<x, y, z> = QQ[]
            sage: S = Spec(R)
            sage: P = S(R.ideal(x, y, z)); P
            Point on Spectrum of Multivariate Polynomial Ring
            in x, y, z over Rational Field defined by the Ideal (x, y, z)
            of Multivariate Polynomial Ring in x, y, z over Rational Field

        This indicates the fix of :trac:`12734`::
            sage: S = Spec(ZZ)
            sage: S(ZZ)
            Set of rational points of Spectrum of Integer Ring
            sage: S(S)
            Set of rational points of Spectrum of Integer Ring
        """
        if is_CommutativeRing(x):
            return self.point_homset(x)
        from sage.schemes.generic.scheme import is_Scheme
        if is_Scheme(x):
            return x.Hom(self)

        return SchemeTopologicalPoint_prime_ideal(self, x)

    def __init__(self, C):
        """
        TESTS::

            sage: from sage.schemes.jacobians.abstract_jacobian import Jacobian_generic
            sage: P2.<x, y, z> = ProjectiveSpace(QQ, 2)
            sage: C = Curve(x^3 + y^3 + z^3)
            sage: J = Jacobian_generic(C); J
            Jacobian of Projective Plane Curve over Rational Field defined by x^3 + y^3 + z^3
            sage: type(J)
            <class 'sage.schemes.jacobians.abstract_jacobian.Jacobian_generic_with_category'>

        Note: this is an abstract parent, so we skip element tests::

            sage: TestSuite(J).run(skip =["_test_an_element",\
                                          "_test_elements",\
                                          "_test_elements_eq_reflexive",\
                                          "_test_elements_eq_symmetric",\
                                          "_test_elements_eq_transitive",\
                                          "_test_elements_neq",\
                                          "_test_some_elements"])

        ::

            sage: Jacobian_generic(ZZ)
            Traceback (most recent call last):
            ...
            TypeError: Argument (=Integer Ring) must be a scheme.
            sage: Jacobian_generic(P2)
            Traceback (most recent call last):
            ...
            ValueError: C (=Projective Space of dimension 2 over Rational Field) must have dimension 1.
            sage: P2.<x, y, z> = ProjectiveSpace(Zmod(6), 2)
            sage: C = Curve(x + y + z)
            sage: Jacobian_generic(C)
            Traceback (most recent call last):
            ...
            TypeError: C (=Projective Plane Curve over Ring of integers modulo 6 defined by x + y + z) must be defined over a field.
        """
        if not is_Scheme(C):
            raise TypeError("Argument (=%s) must be a scheme."%C)
        if C.base_ring() not in _Fields:
            raise TypeError("C (=%s) must be defined over a field."%C)
        if C.dimension() != 1:
            raise ValueError("C (=%s) must have dimension 1."%C)
        self.__curve = C
        Scheme.__init__(self, C.base_scheme())

def Schemes(X=None):
    """
    Construct a category of schemes.

    EXAMPLES::

        sage: Schemes()
        Category of Schemes

        sage: Schemes(Spec(ZZ))
        Category of schemes over Integer Ring

        sage: Schemes(ZZ)
        Category of schemes over Integer Ring
    """
    if X is None:
        return Schemes_abstract()
    from sage.schemes.generic.scheme import is_Scheme
    if not is_Scheme(X):
        X = Schemes()(X)
    return Schemes_over_base(X)

    def __classcall_private__(cls, X = None):
        """
        Implement the dispatching ``Schemes(ZZ)`` -> ``Schemes_over_base``.

        EXAMPLES::

            sage: Schemes()
            Category of schemes

            sage: Schemes(Spec(ZZ))
            Category of schemes over Integer Ring

            sage: Schemes(ZZ)
            Category of schemes over Integer Ring
        """
        if X is not None:
            from sage.schemes.generic.scheme import is_Scheme
            if not is_Scheme(X):
                X = Schemes()(X)
            return Schemes_over_base(X)
        else:
            return super(Schemes, cls).__classcall__(cls)

    def _call_(self, x):
        """
        Construct a scheme from the data in ``x``

        EXAMPLES:

        Let us first construct the category of schemes::

            sage: S = Schemes(); S
            Category of schemes

        We create a scheme from a ring::

            sage: X = S(ZZ); X                  # indirect doctest
            Spectrum of Integer Ring

        We create a scheme from a scheme (do nothing)::

            sage: S(X)
            Spectrum of Integer Ring

        We create a scheme morphism from a ring homomorphism.x::

            sage: phi = ZZ.hom(QQ); phi
            Ring Coercion morphism:
              From: Integer Ring
              To:   Rational Field
            sage: f = S(phi); f                 # indirect doctest
            Affine Scheme morphism:
              From: Spectrum of Rational Field
              To:   Spectrum of Integer Ring
              Defn: Ring Coercion morphism:
                      From: Integer Ring
                      To:   Rational Field

            sage: f.domain()
            Spectrum of Rational Field
            sage: f.codomain()
            Spectrum of Integer Ring
            sage: S(f)                          # indirect doctest
            Affine Scheme morphism:
              From: Spectrum of Rational Field
              To:   Spectrum of Integer Ring
              Defn: Ring Coercion morphism:
                      From: Integer Ring
                      To:   Rational Field

        """
        from sage.schemes.generic.scheme import is_Scheme
        if is_Scheme(x):
            return x
        from sage.schemes.generic.morphism import is_SchemeMorphism
        if is_SchemeMorphism(x):
            return x
        from sage.rings.morphism import is_RingHomomorphism
        from sage.rings.ring import CommutativeRing
        from sage.schemes.generic.spec import Spec
        if isinstance(x, CommutativeRing):
            return Spec(x)
        elif is_RingHomomorphism(x):
            A = Spec(x.codomain())
            return A.hom(x)
        else:
            raise TypeError("No way to create an object or morphism in %s from %s"%(self, x))

def enum_projective_rational_field(X,B):
    r"""
    Enumerates projective, rational points on scheme ``X`` of height up to
    bound ``B``.

    INPUT:

    - ``X`` -  a scheme or set of abstract rational points of a scheme;
    - ``B`` -  a positive integer bound.

    OUTPUT:

    - a list containing the projective points of ``X`` of height up to ``B``,
      sorted.

    EXAMPLES::

        sage: P.<X,Y,Z> = ProjectiveSpace(2,QQ)
        sage: C = P.subscheme([X+Y-Z])
        sage: from sage.schemes.projective.projective_rational_point import enum_projective_rational_field
        sage: enum_projective_rational_field(C(QQ),6)
        [(-5 : 6 : 1), (-4 : 5 : 1), (-3 : 4 : 1), (-2 : 3 : 1),
         (-3/2 : 5/2 : 1), (-1 : 1 : 0), (-1 : 2 : 1), (-2/3 : 5/3 : 1),
         (-1/2 : 3/2 : 1), (-1/3 : 4/3 : 1), (-1/4 : 5/4 : 1),
         (-1/5 : 6/5 : 1), (0 : 1 : 1), (1/6 : 5/6 : 1), (1/5 : 4/5 : 1),
         (1/4 : 3/4 : 1), (1/3 : 2/3 : 1), (2/5 : 3/5 : 1), (1/2 : 1/2 : 1),
         (3/5 : 2/5 : 1), (2/3 : 1/3 : 1), (3/4 : 1/4 : 1), (4/5 : 1/5 : 1),
         (5/6 : 1/6 : 1), (1 : 0 : 1), (6/5 : -1/5 : 1), (5/4 : -1/4 : 1),
         (4/3 : -1/3 : 1), (3/2 : -1/2 : 1), (5/3 : -2/3 : 1), (2 : -1 : 1),
         (5/2 : -3/2 : 1), (3 : -2 : 1), (4 : -3 : 1), (5 : -4 : 1),
         (6 : -5 : 1)]
        sage: enum_projective_rational_field(C,6) == enum_projective_rational_field(C(QQ),6)
        True

    ::

        sage: P3.<W,X,Y,Z> = ProjectiveSpace(3,QQ)
        sage: enum_projective_rational_field(P3,1)
        [(-1 : -1 : -1 : 1), (-1 : -1 : 0 : 1), (-1 : -1 : 1 : 0), (-1 : -1 : 1 : 1),
        (-1 : 0 : -1 : 1), (-1 : 0 : 0 : 1), (-1 : 0 : 1 : 0), (-1 : 0 : 1 : 1),
        (-1 : 1 : -1 : 1), (-1 : 1 : 0 : 0), (-1 : 1 : 0 : 1), (-1 : 1 : 1 : 0),
        (-1 : 1 : 1 : 1), (0 : -1 : -1 : 1), (0 : -1 : 0 : 1), (0 : -1 : 1 : 0),
        (0 : -1 : 1 : 1), (0 : 0 : -1 : 1), (0 : 0 : 0 : 1), (0 : 0 : 1 : 0),
        (0 : 0 : 1 : 1), (0 : 1 : -1 : 1), (0 : 1 : 0 : 0), (0 : 1 : 0 : 1),
        (0 : 1 : 1 : 0), (0 : 1 : 1 : 1), (1 : -1 : -1 : 1), (1 : -1 : 0 : 1),
        (1 : -1 : 1 : 0), (1 : -1 : 1 : 1), (1 : 0 : -1 : 1), (1 : 0 : 0 : 0),
        (1 : 0 : 0 : 1), (1 : 0 : 1 : 0), (1 : 0 : 1 : 1), (1 : 1 : -1 : 1),
        (1 : 1 : 0 : 0), (1 : 1 : 0 : 1), (1 : 1 : 1 : 0), (1 : 1 : 1 : 1)]

    ALGORITHM:

    We just check all possible projective points in correct dimension
    of projective space to see if they lie on ``X``.

    AUTHORS:

    - John Cremona and Charlie Turner (06-2010)
    """
    from sage.schemes.projective.projective_space import is_ProjectiveSpace
    if(is_Scheme(X)):
        if (not is_ProjectiveSpace(X.ambient_space())):
            raise TypeError("Ambient space must be projective space over the rational field")
        X = X(X.base_ring())
    else:
        if (not is_ProjectiveSpace(X.codomain().ambient_space())):
            raise TypeError("Codomain must be projective space over the rational field")

    n = X.codomain().ambient_space().ngens()
    zero = (0,) * n
    pts = []
    for c in cartesian_product_iterator([srange(-B,B+1) for _ in range(n)]):
        if gcd(c) == 1 and c > zero:
            try:
                pts.append(X(c))
            except TypeError:
                pass
    pts.sort()
    return pts

def enum_projective_finite_field(X):
    """
    Enumerates projective points on scheme ``X`` defined over a finite field.

    INPUT:

    - ``X`` -  a scheme defined over a finite field or a set of abstract
      rational points of such a scheme.

    OUTPUT:

    - a list containing the projective points of ``X`` over the finite field,
      sorted.

    EXAMPLES::

        sage: F = GF(53)
        sage: P.<X,Y,Z> = ProjectiveSpace(2,F)
        sage: from sage.schemes.projective.projective_rational_point import enum_projective_finite_field
        sage: len(enum_projective_finite_field(P(F)))
        2863
        sage: 53^2+53+1
        2863

    ::

        sage: F = GF(9,'a')
        sage: P.<X,Y,Z> = ProjectiveSpace(2,F)
        sage: C = Curve(X^3-Y^3+Z^2*Y)
        sage: enum_projective_finite_field(C(F))
        [(0 : 0 : 1), (0 : 1 : 1), (0 : 2 : 1), (1 : 1 : 0), (a + 1 : 2*a : 1),
        (a + 1 : 2*a + 1 : 1), (a + 1 : 2*a + 2 : 1), (2*a + 2 : a : 1),
        (2*a + 2 : a + 1 : 1), (2*a + 2 : a + 2 : 1)]

    ::

        sage: F = GF(5)
        sage: P2F.<X,Y,Z> = ProjectiveSpace(2,F)
        sage: enum_projective_finite_field(P2F)
        [(0 : 0 : 1), (0 : 1 : 0), (0 : 1 : 1), (0 : 2 : 1), (0 : 3 : 1), (0 : 4 : 1),
        (1 : 0 : 0), (1 : 0 : 1), (1 : 1 : 0), (1 : 1 : 1), (1 : 2 : 1), (1 : 3 : 1),
        (1 : 4 : 1), (2 : 0 : 1), (2 : 1 : 0), (2 : 1 : 1), (2 : 2 : 1), (2 : 3 : 1),
        (2 : 4 : 1), (3 : 0 : 1), (3 : 1 : 0), (3 : 1 : 1), (3 : 2 : 1), (3 : 3 : 1),
        (3 : 4 : 1), (4 : 0 : 1), (4 : 1 : 0), (4 : 1 : 1), (4 : 2 : 1), (4 : 3 : 1),
        (4 : 4 : 1)]

    ALGORITHM:

    Checks all points in projective space to see if they lie on X.

    .. WARNING::

        If ``X`` is defined over an infinite field, this code will not finish!

    AUTHORS:

    - John Cremona and Charlie Turner (06-2010).
    """
    from sage.schemes.projective.projective_space import is_ProjectiveSpace
    if(is_Scheme(X)):
        if (not is_ProjectiveSpace(X.ambient_space())):
            raise TypeError("Ambient space must be projective space over a finite")
        X = X(X.base_ring())
    else:
        if (not is_ProjectiveSpace(X.codomain().ambient_space())):
            raise TypeError("Codomain must be projective space over a finite field")

    n = X.codomain().ambient_space().ngens()-1
    F = X.value_ring()
    pts = []
    for k in range(n+1):
        for c in cartesian_product_iterator([F for _ in range(k)]):
            try:
                pts.append(X(list(c)+[1]+[0]*(n-k)))
            except TypeError:
                pass
    pts.sort()
    return pts

def enum_affine_rational_field(X,B):
    """
    Enumerates affine rational points on scheme ``X`` (defined over `\QQ`) up
    to bound ``B``.

    INPUT:

    - ``X`` -  a scheme or set of abstract rational points of a scheme;
    - ``B`` -  a positive integer bound.

    OUTPUT:

    - a list containing the affine points of ``X`` of height up to ``B``,
      sorted.

    EXAMPLES::

        sage: A.<x,y,z> = AffineSpace(3,QQ)
        sage: from sage.schemes.affine.affine_rational_point import enum_affine_rational_field
        sage: enum_affine_rational_field(A(QQ),1)
        [(-1, -1, -1), (-1, -1, 0), (-1, -1, 1), (-1, 0, -1), (-1, 0, 0), (-1, 0, 1),
        (-1, 1, -1), (-1, 1, 0), (-1, 1, 1), (0, -1, -1), (0, -1, 0), (0, -1, 1),
        (0, 0, -1), (0, 0, 0), (0, 0, 1), (0, 1, -1), (0, 1, 0), (0, 1, 1), (1, -1, -1),
        (1, -1, 0), (1, -1, 1), (1, 0, -1), (1, 0, 0), (1, 0, 1), (1, 1, -1), (1, 1, 0),
        (1, 1, 1)]

    ::

        sage: A.<w,x,y,z> = AffineSpace(4,QQ)
        sage: S = A.subscheme([x^2-y*z+3,w^3+z+y^2])
        sage: enum_affine_rational_field(S(QQ),2)
        []
        sage: enum_affine_rational_field(S(QQ),3)
        [(-2, 0, -3, -1)]

    ::

        sage: A.<x,y> = AffineSpace(2,QQ)
        sage: C = Curve(x^2+y-x)
        sage: enum_affine_rational_field(C,10)
        [(-2, -6), (-1, -2), (0, 0), (1, 0), (2, -2), (3, -6)]


    AUTHORS:

    - David R. Kohel <[email protected]>: original version.

    - Charlie Turner (06-2010): small adjustments.
    """
    if is_Scheme(X):
        X = X(X.base_ring())
    n = X.codomain().ambient_space().ngens()
    if X.value_ring() is ZZ:
        Q = [ 1 ]
    else: # rational field
        Q = range(1, B + 1)
    R = [ 0 ] + [ s*k for k in range(1, B+1) for s in [1, -1] ]
    pts = []
    P = [0] * n
    m = ZZ(0)
    try:
        pts.append(X(P))
    except TypeError:
        pass
    iters = [ iter(R) for _ in range(n) ]
    for it in iters:
        it.next()
    i = 0
    while i < n:
        try:
            a = ZZ(iters[i].next())
        except StopIteration:
            iters[i] = iter(R) # reset
            P[i] = iters[i].next() # reset P[i] to 0 and increment
            i += 1
            continue
        m = m.gcd(a)
        P[i] = a
        for b in Q:
            if m.gcd(b) == 1:
                try:
                    pts.append(X([ num/b for num in P ]))
                except TypeError:
                    pass
        i = 0
        m = ZZ(0)
    pts.sort()
    return pts

def enum_affine_finite_field(X):
    r"""
    Enumerates affine points on scheme ``X`` defined over a finite field.

    INPUT:

    - ``X`` -  a scheme defined over a finite field or a set of abstract
      rational points of such a scheme.

    OUTPUT:

    - a list containing the affine points of ``X`` over the finite field,
      sorted.

    EXAMPLES::

        sage: F = GF(7)
        sage: A.<w,x,y,z> = AffineSpace(4,F)
        sage: C = A.subscheme([w^2+x+4,y*z*x-6,z*y+w*x])
        sage: from sage.schemes.affine.affine_rational_point import enum_affine_finite_field
        sage: enum_affine_finite_field(C(F))
        []
        sage: C = A.subscheme([w^2+x+4,y*z*x-6])
        sage: enum_affine_finite_field(C(F))
        [(0, 3, 1, 2), (0, 3, 2, 1), (0, 3, 3, 3), (0, 3, 4, 4), (0, 3, 5, 6),
        (0, 3, 6, 5), (1, 2, 1, 3), (1, 2, 2, 5), (1, 2, 3, 1), (1, 2, 4, 6),
        (1, 2, 5, 2), (1, 2, 6, 4), (2, 6, 1, 1), (2, 6, 2, 4), (2, 6, 3, 5),
        (2, 6, 4, 2), (2, 6, 5, 3), (2, 6, 6, 6), (3, 1, 1, 6), (3, 1, 2, 3),
        (3, 1, 3, 2), (3, 1, 4, 5), (3, 1, 5, 4), (3, 1, 6, 1), (4, 1, 1, 6),
        (4, 1, 2, 3), (4, 1, 3, 2), (4, 1, 4, 5), (4, 1, 5, 4), (4, 1, 6, 1),
        (5, 6, 1, 1), (5, 6, 2, 4), (5, 6, 3, 5), (5, 6, 4, 2), (5, 6, 5, 3),
        (5, 6, 6, 6), (6, 2, 1, 3), (6, 2, 2, 5), (6, 2, 3, 1), (6, 2, 4, 6),
        (6, 2, 5, 2), (6, 2, 6, 4)]

    ::

        sage: A.<x,y,z> = AffineSpace(3,GF(3))
        sage: S = A.subscheme(x+y)
        sage: enum_affine_finite_field(S)
        [(0, 0, 0), (0, 0, 1), (0, 0, 2), (1, 2, 0), (1, 2, 1), (1, 2, 2),
        (2, 1, 0), (2, 1, 1), (2, 1, 2)]

    ALGORITHM:

    Checks all points in affine space to see if they lie on X.

    .. WARNING::

        If ``X`` is defined over an infinite field, this code will not finish!

    AUTHORS:

    - John Cremona and Charlie Turner (06-2010)
    """
    if is_Scheme(X):
        X = X(X.base_ring())
    n = X.codomain().ambient_space().ngens()
    F = X.value_ring()
    pts = []
    for c in cartesian_product_iterator([F]*n):
        try:
            pts.append(X(c))
        except Exception:
            pass
    pts.sort()
    return pts

def enum_projective_number_field(X,B, prec=53):
    """
    Enumerates projective points on scheme ``X`` defined over a number field. Simply checks all of the
    points of absolute height of at most ``B`` and adds those that are on the scheme to the list.

    INPUT:

    - ``X`` - a scheme defined over a number field

    - ``B`` - a real number

    - ``prec`` - the precision to use for computing the elements of bounded height of number fields

    OUTPUT:

     - a list containing the projective points of ``X`` of absolute height up to ``B``,
       sorted.

    .. WARNING::

       In the current implementation, the output of the [Doyle-Krumm] algorithm
       for elements of bounded height cannot be guaranteed to be correct due to
       the necessity of floating point computations. In some cases, the default
       53-bit precision is considerably lower than would be required for the
       algorithm to generate correct output.

    EXAMPLES::

        sage: from sage.schemes.projective.projective_rational_point import enum_projective_number_field
        sage: u = QQ['u'].0
        sage: K = NumberField(u^3 - 5,'v')
        sage: P.<x,y,z> = ProjectiveSpace(K, 2)
        sage: X = P.subscheme([x - y])
        sage: enum_projective_number_field(X(K), 5^(1/3), prec=2^10)
        [(0 : 0 : 1), (-1 : -1 : 1), (1 : 1 : 1), (-1/5*v^2 : -1/5*v^2 : 1), (-v : -v : 1),
        (1/5*v^2 : 1/5*v^2 : 1), (v : v : 1), (1 : 1 : 0)]

    ::

        sage: u = QQ['u'].0
        sage: K = NumberField(u^2 + 3, 'v')
        sage: A.<x,y> = ProjectiveSpace(K,1)
        sage: X=A.subscheme(x-y)
        sage: from sage.schemes.projective.projective_rational_point import enum_projective_number_field
        sage: enum_projective_number_field(X, 2)
        [(1 : 1)]
    """
    from sage.schemes.projective.projective_space import is_ProjectiveSpace
    if(is_Scheme(X)):
        if (not is_ProjectiveSpace(X.ambient_space())):
            raise TypeError("Ambient space must be projective space over a number field")
        X = X(X.base_ring())
    else:
        if (not is_ProjectiveSpace(X.codomain().ambient_space())):
            raise TypeError("Codomain must be projective space over a number field")

    R = X.codomain().ambient_space()

    pts = []

    for P in R.points_of_bounded_height(B, prec):
        try:
            pts.append(X(P))
        except TypeError:
            pass
    pts.sort()
    return pts