Python ell_generic.is_EllipticCurve函数代码示例

    def point(self, v, check=True):
        """
        Create a point.

        INPUT:

        - ``v`` -- anything that defines a point.

        - ``check`` -- boolean (optional, default=``True``). Whether
          to check the defining data for consistency.

        OUTPUT:

        A point of the scheme.

        EXAMPLES::

            sage: A2 = AffineSpace(QQ,2)
            sage: A2.point([4,5])
            (4, 5)

            sage: R.<t> = PolynomialRing(QQ)
            sage: E = EllipticCurve([t + 1, t, t, 0, 0])
            sage: E.point([0, 0])
            (0 : 0 : 1)
        """
        # todo: update elliptic curve stuff to take point_homset as argument
        from sage.schemes.elliptic_curves.ell_generic import is_EllipticCurve
        if is_EllipticCurve(self):
            try:
                return self._point(self.point_homset(), v, check=check)
            except AttributeError:  # legacy code without point_homset
                return self._point(self, v, check=check)

        return self.point_homset() (v, check=check)

    def point(self, v, check=True):
        """
        Create a point.

        INPUT:
        
        - ``v`` -- anything that defines a point.

        - ``check`` -- boolean (optional, default=``True``). Whether
          to check the defining data for consistency.

        OUTPUT:

        A point of the scheme.

        EXAMPLES::
        
            sage: A2 = AffineSpace(QQ,2)
            sage: A2.point([4,5])
            (4, 5)
        """
        # todo: update elliptic curve stuff to take point_homset as argument
        from sage.schemes.elliptic_curves.ell_generic import is_EllipticCurve
        if is_EllipticCurve(self):
            return self._point(self, v, check=check)

        return self.point_homset() (v, check=check)

    def point(self, v, check=True):
        """
        Create a point on this projective subscheme.

        INPUT:

        - ``v`` -- anything that defines a point

        - ``check`` -- boolean (optional, default: ``True``); whether
          to check the defining data for consistency

        OUTPUT: A point of the subscheme.

        EXAMPLES::

            sage: P2.<x,y,z> = ProjectiveSpace(QQ, 2)
            sage: X = P2.subscheme([x-y,y-z])
            sage: X.point([1,1,1])
            (1 : 1 : 1)

        ::

            sage: P2.<x,y> = ProjectiveSpace(QQ, 1)
            sage: X = P2.subscheme([y])
            sage: X.point(infinity)
            (1 : 0)

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ, 1)
            sage: X = P.subscheme(x^2+2*y^2)
            sage: X.point(infinity)
            Traceback (most recent call last):
            ...
            TypeError: Coordinates [1, 0] do not define a point on Closed subscheme
            of Projective Space of dimension 1 over Rational Field defined by:
              x^2 + 2*y^2
        """
        from sage.rings.infinity import infinity
        if v is infinity  or\
          (isinstance(v, (list,tuple)) and len(v) == 1 and v[0] is infinity):
            if self.ambient_space().dimension_relative() > 1:
                raise ValueError("%s not well defined in dimension > 1"%v)
            v = [1, 0]
        # todo: update elliptic curve stuff to take point_homset as argument
        from sage.schemes.elliptic_curves.ell_generic import is_EllipticCurve
        if is_EllipticCurve(self):
            try:
                return self._point(self.point_homset(), v, check=check)
            except AttributeError:  # legacy code without point_homset
                return self._point(self, v, check=check)

        return self.point_homset()(v, check=check)

    def is_isogenous(self, other, field=None):
        """
        Returns whether or not self is isogenous to other.

        INPUT:

        - ``other`` -- another elliptic curve.

        - ``field`` (default None) -- Currently not implemented. A
          field containing the base fields of the two elliptic curves
          onto which the two curves may be extended to test if they
          are isogenous over this field. By default is_isogenous will
          not try to find this field unless one of the curves can be
          be extended into the base field of the other, in which case
          it will test over the larger base field.

        OUTPUT:

        (bool) True if there is an isogeny from curve ``self`` to
        curve ``other`` defined over ``field``.

        METHOD:

        Over general fields this is only implemented in trivial cases.

        EXAMPLES::

            sage: E1 = EllipticCurve(CC, [1,18]); E1
            Elliptic Curve defined by y^2 = x^3 + 1.00000000000000*x + 18.0000000000000 over Complex Field with 53 bits of precision
            sage: E2 = EllipticCurve(CC, [2,7]); E2
            Elliptic Curve defined by y^2 = x^3 + 2.00000000000000*x + 7.00000000000000 over Complex Field with 53 bits of precision
            sage: E1.is_isogenous(E2)
            Traceback (most recent call last):
            ...
            NotImplementedError: Only implemented for isomorphic curves over general fields.

            sage: E1 = EllipticCurve(Frac(PolynomialRing(ZZ,'t')), [2,19]); E1
            Elliptic Curve defined by y^2 = x^3 + 2*x + 19 over Fraction Field of Univariate Polynomial Ring in t over Integer Ring
            sage: E2 = EllipticCurve(CC, [23,4]); E2
            Elliptic Curve defined by y^2 = x^3 + 23.0000000000000*x + 4.00000000000000 over Complex Field with 53 bits of precision
            sage: E1.is_isogenous(E2)
            Traceback (most recent call last):
            ...
            NotImplementedError: Only implemented for isomorphic curves over general fields.
        """
        from ell_generic import is_EllipticCurve
        if not is_EllipticCurve(other):
            raise ValueError("Second argument is not an Elliptic Curve.")
        if self.is_isomorphic(other):
            return True
        else:
            raise NotImplementedError("Only implemented for isomorphic curves over general fields.")

    def is_sextic_twist(self, other):
        r"""
        Determine whether this curve is a sextic twist of another.

        INPUT:

        - ``other`` -- an elliptic curves with the same base field as self.

        OUTPUT:

        Either 0, if the curves are not sextic twists, or `D` if
        ``other`` is ``self.sextic_twist(D)`` (up to isomorphism).
        If ``self`` and ``other`` are isomorphic, returns 1.

        .. note::

           Not fully implemented in characteristics 2 or 3.

        EXAMPLES::

            sage: E = EllipticCurve_from_j(GF(13)(0))
            sage: E1 = E.sextic_twist(2)
            sage: D = E.is_sextic_twist(E1); D!=0
            True
            sage: E.sextic_twist(D).is_isomorphic(E1)
            True

        ::

            sage: E = EllipticCurve_from_j(0)
            sage: E1 = E.sextic_twist(12345)
            sage: D = E.is_sextic_twist(E1); D
            575968320
            sage: (D/12345).is_perfect_power(6)
            True
        """
        from sage.schemes.elliptic_curves.ell_generic import is_EllipticCurve
        E = self
        F = other
        if not is_EllipticCurve(E) or not is_EllipticCurve(F):
            raise ValueError("arguments are not elliptic curves")
        K = E.base_ring()
        zero = K.zero_element()
        if not K == F.base_ring():
            return zero
        j=E.j_invariant()
        if  j != F.j_invariant() or not j.is_zero():
            return zero

        if E.is_isomorphic(F):
            return K.one_element()

        char=K.characteristic()

        if char==2:
            raise NotImplementedError("not implemented in characteristic 2")
        elif char==3:
            raise NotImplementedError("not implemented in characteristic 3")
        else:
            # now char!=2,3:
            D = F.c6()/E.c6()

        if D.is_zero():
            return D

        assert E.sextic_twist(D).is_isomorphic(F)

        return D

    def is_quadratic_twist(self, other):
        r"""
        Determine whether this curve is a quadratic twist of another.

        INPUT:

        - ``other`` -- an elliptic curves with the same base field as self.

        OUTPUT:

        Either 0, if the curves are not quadratic twists, or `D` if
        ``other`` is ``self.quadratic_twist(D)`` (up to isomorphism).
        If ``self`` and ``other`` are isomorphic, returns 1.

        If the curves are defined over `\mathbb{Q}`, the output `D` is
        a squarefree integer.

        .. note::

           Not fully implemented in characteristic 2, or in
           characteristic 3 when both `j`-invariants are 0.

        EXAMPLES::

            sage: E = EllipticCurve('11a1')
            sage: Et = E.quadratic_twist(-24)
            sage: E.is_quadratic_twist(Et)
            -6

            sage: E1=EllipticCurve([0,0,1,0,0])
            sage: E1.j_invariant()
            0
            sage: E2=EllipticCurve([0,0,0,0,2])
            sage: E1.is_quadratic_twist(E2)
            2
            sage: E1.is_quadratic_twist(E1)
            1
            sage: type(E1.is_quadratic_twist(E1)) == type(E1.is_quadratic_twist(E2))   #trac 6574
            True

        ::

            sage: E1=EllipticCurve([0,0,0,1,0])
            sage: E1.j_invariant()
            1728
            sage: E2=EllipticCurve([0,0,0,2,0])
            sage: E1.is_quadratic_twist(E2)
            0
            sage: E2=EllipticCurve([0,0,0,25,0])
            sage: E1.is_quadratic_twist(E2)
            5

        ::

            sage: F = GF(101)
            sage: E1 = EllipticCurve(F,[4,7])
            sage: E2 = E1.quadratic_twist()
            sage: D = E1.is_quadratic_twist(E2); D!=0
            True
            sage: F = GF(101)
            sage: E1 = EllipticCurve(F,[4,7])
            sage: E2 = E1.quadratic_twist()
            sage: D = E1.is_quadratic_twist(E2)
            sage: E1.quadratic_twist(D).is_isomorphic(E2)
            True
            sage: E1.is_isomorphic(E2)
            False
            sage: F2 = GF(101^2,'a')
            sage: E1.change_ring(F2).is_isomorphic(E2.change_ring(F2))
            True

        A characteristic 3 example::

            sage: F = GF(3^5,'a')
            sage: E1 = EllipticCurve_from_j(F(1))
            sage: E2 = E1.quadratic_twist(-1)
            sage: D = E1.is_quadratic_twist(E2); D!=0
            True
            sage: E1.quadratic_twist(D).is_isomorphic(E2)
            True

        ::

            sage: E1 = EllipticCurve_from_j(F(0))
            sage: E2 = E1.quadratic_twist()
            sage: D = E1.is_quadratic_twist(E2); D
            1
            sage: E1.is_isomorphic(E2)
            True

        """
        from sage.schemes.elliptic_curves.ell_generic import is_EllipticCurve
        E = self
        F = other
        if not is_EllipticCurve(E) or not is_EllipticCurve(F):
            raise ValueError("arguments are not elliptic curves")
        K = E.base_ring()
        zero = K.zero_element()
        if not K == F.base_ring():
            return zero
#.........这里部分代码省略.........