Coverage for local/lib/python2.7/site-packages/sage/schemes/hyperelliptic_curves/jacobian_homset.py : 51%

""" 

Rational point sets on a Jacobian 

EXAMPLES:: 

sage: x = QQ['x'].0 

sage: f = x^5 + x + 1 

sage: C = HyperellipticCurve(f); C 

Hyperelliptic Curve over Rational Field defined by y^2 = x^5 + x + 1 

sage: C(QQ) 

Set of rational points of Hyperelliptic Curve over Rational Field defined by y^2 = x^5 + x + 1 

sage: P = C([0,1,1]) 

sage: J = C.jacobian(); J 

Jacobian of Hyperelliptic Curve over Rational Field defined by y^2 = x^5 + x + 1 

sage: Q = J(QQ)(P); Q 

(x, y - 1) 

sage: Q + Q 

(x^2, y - 1/2*x - 1) 

sage: Q*3 

(x^2 - 1/64*x + 1/8, y + 255/512*x + 65/64) 

:: 

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) 

""" 

#***************************************************************************** 

# 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 absolute_import 

from six import integer_types 

from sage.rings.all import PolynomialRing, Integer, ZZ 

from sage.rings.integer import is_Integer 

from sage.rings.polynomial.polynomial_element import is_Polynomial 

from sage.schemes.generic.homset import SchemeHomset_points 

from sage.schemes.generic.morphism import is_SchemeMorphism 

from .jacobian_morphism import JacobianMorphism_divisor_class_field 

class JacobianHomset_divisor_classes(SchemeHomset_points): 

def __init__(self, Y, X, **kwds): 

R = X.base_ring() 

S = Y.coordinate_ring() 

SchemeHomset_points.__init__(self, Y, X, **kwds) 

P2 = X.curve()._printing_ring 

if S != R: 

y = str(P2.gen()) 

x = str(P2.base_ring().gen()) 

P1 = PolynomialRing(S,name=x) 

P2 = PolynomialRing(P1,name=y) 

self._printing_ring = P2 

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, integer_types + (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 _cmp_(self,other): 

if self.curve() == other.curve(): 

return 0 

else: 

return -1 

def _morphism(self, *args, **kwds): 

return JacobianMorphism_divisor_class_field(*args, **kwds) 

def curve(self): 

return self.codomain().curve() 

def value_ring(self): 

""" 

Returns S for a homset X(T) where T = Spec(S). 

""" 

from sage.schemes.generic.scheme import is_AffineScheme 

T = self.domain() 

if is_AffineScheme(T): 

return T.coordinate_ring() 

else: 

raise TypeError("Domain of argument must be of the form Spec(S).") 

def base_extend(self, R): 

if R != ZZ: 

raise NotImplementedError("Jacobian point sets viewed as modules over rings other than ZZ not implemented") 

return self