text: """
  Jacobian of a general hyperelliptic curve
  """
  
  #*****************************************************************************
  #  Copyright (C) 2006 David Kohel <kohel@maths.usyd.edu>
  #  Distributed under the terms of the GNU General Public License (GPL)
  #                  http://www.gnu.org/licenses/
  #*****************************************************************************
  from __future__ import print_function
  from __future__ import absolute_import
  
  from sage.rings.all import Integer
  from sage.schemes.jacobians.abstract_jacobian import Jacobian_generic
  from . import jacobian_homset
  from . import jacobian_morphism
  
  class HyperellipticJacobian_generic(Jacobian_generic):
      """
      EXAMPLES::
  
          sage: FF = FiniteField(2003)
          sage: R.<x> = PolynomialRing(FF)
          sage: f = x**5 + 1184*x**3 + 1846*x**2 + 956*x + 560
          sage: C = HyperellipticCurve(f)
          sage: J = C.jacobian()
          sage: a = x**2 + 376*x + 245; b = 1015*x + 1368
          sage: X = J(FF)
          sage: D = X([a,b])
          sage: D
          (x^2 + 376*x + 245, y + 988*x + 635)
          sage: J(0)
          (1)
          sage: D == J([a,b])
          True
          sage: D == D + J(0)
          True
  
      An more extended example, demonstrating arithmetic in J(QQ) and
      J(K) for a number field K/QQ.
  
      ::
  
          sage: P.<x> = PolynomialRing(QQ)
          sage: f = x^5 - x + 1; h = x
          sage: C = HyperellipticCurve(f,h,'u,v')
          sage: C
          Hyperelliptic Curve over Rational Field defined by v^2 + u*v = u^5 - u + 1
          sage: PP = C.ambient_space()
          sage: PP
          Projective Space of dimension 2 over Rational Field
          sage: C.defining_polynomial()
          -x0^5 + x0*x1*x2^3 + x1^2*x2^3 + x0*x2^4 - x2^5
          sage: C(QQ)
          Set of rational points of Hyperelliptic Curve over Rational Field defined by v^2 + u*v = u^5 - u + 1
          sage: K.<t> = NumberField(x^2-2)
          sage: C(K)
          Set of rational points of Hyperelliptic Curve over Number Field in t with defining polynomial x^2 - 2 defined by v^2 + u*v = u^5 - u + 1
          sage: P = C(QQ)(0,1,1); P
          (0 : 1 : 1)
          sage: P == C(0,1,1)
          True
          sage: C(0,1,1).parent()
          Set of rational points of Hyperelliptic Curve over Rational Field defined by v^2 + u*v = u^5 - u + 1
          sage: P1 = C(K)(P)
          sage: P2 = C(K)([2,4*t-1,1])
          sage: P3 = C(K)([-1/2,1/8*(7*t+2),1])
          sage: P1, P2, P3
          ((0 : 1 : 1), (2 : 4*t - 1 : 1), (-1/2 : 7/8*t + 1/4 : 1))
          sage: J = C.jacobian()
          sage: J
          Jacobian of Hyperelliptic Curve over Rational Field defined by v^2 + u*v = u^5 - u + 1
          sage: Q = J(QQ)(P); Q
          (u, v - 1)
          sage: for i in range(6): Q*i
          (1)
          (u, v - 1)
          (u^2, v + u - 1)
          (u^2, v + 1)
          (u, v + 1)
          (1)
          sage: Q1 = J(K)(P1); print("%s -> %s"%( P1, Q1 ))
          (0 : 1 : 1) -> (u, v - 1)
          sage: Q2 = J(K)(P2); print("%s -> %s"%( P2, Q2 ))
          (2 : 4*t - 1 : 1) -> (u - 2, v - 4*t + 1)
          sage: Q3 = J(K)(P3); print("%s -> %s"%( P3, Q3 ))
          (-1/2 : 7/8*t + 1/4 : 1) -> (u + 1/2, v - 7/8*t - 1/4)
          sage: R.<x> = PolynomialRing(K)
          sage: Q4 = J(K)([x^2-t,R(1)])
          sage: for i in range(4): Q4*i
          (1)
          (u^2 - t, v - 1)
          (u^2 + (-3/4*t - 9/16)*u + 1/2*t + 1/4, v + (-1/32*t - 57/64)*u + 1/2*t + 9/16)
          (u^2 + (1352416/247009*t - 1636930/247009)*u - 1156544/247009*t + 1900544/247009, v + (-2326345442/122763473*t + 3233153137/122763473)*u + 2439343104/122763473*t - 3350862929/122763473)
          sage: R2 = Q2*5; R2
          (u^2 - 3789465233/116983808*u - 267915823/58491904, v + (-233827256513849/1789384327168*t + 1/2)*u - 15782925357447/894692163584*t)
          sage: R3 = Q3*5; R3
          (u^2 + 5663300808399913890623/14426454798950909645952*u - 26531814176395676231273/28852909597901819291904, v + (253155440321645614070860868199103/2450498420175733688903836378159104*t + 1/2)*u + 2427708505064902611513563431764311/4900996840351467377807672756318208*t)
          sage: R4 = Q4*5; R4
          (u^2 - 3789465233/116983808*u - 267915823/58491904, v + (233827256513849/1789384327168*t + 1/2)*u + 15782925357447/894692163584*t)
          sage: # Thus we find the following identity:
          sage: 5*Q2 + 5*Q4
          (1)
          sage: # Moreover the following relation holds in the 5-torsion subgroup:
          sage: Q2 + Q4 == 2*Q1
          True
      """
  
      def dimension(self):
          """
          Return the dimension of this Jacobian.
  
          OUTPUT:
              Integer
  
          EXAMPLES::
  
              sage: k.<a> = GF(9); R.<x> = k[]
              sage: HyperellipticCurve(x^3 + x - 1, x+a).jacobian().dimension()
              1
              sage: g = HyperellipticCurve(x^6 + x - 1, x+a).jacobian().dimension(); g
              2
              sage: type(g)
              <... 'sage.rings.integer.Integer'>
          """
          return Integer(self.curve().genus())
  
      def point(self, mumford, check=True):
          try:
              return self(self.base_ring())(mumford)
          except AttributeError:
              raise ValueError("Arguments must determine a valid Mumford divisor.")
  
      def _point_homset(self, *args, **kwds):
          return jacobian_homset.JacobianHomset_divisor_classes(*args, **kwds)
  
      def _point(self, *args, **kwds):
          return jacobian_morphism.JacobianMorphism_divisor_class_field(*args, **kwds)
  
      def _cmp_(self,other):
          if self.curve() == other.curve():
              return 0
          else:
              return -1
  
  