Coverage for local/lib/python2.7/site-packages/sage/schemes/hyperelliptic_curves/jacobian_generic.py : 18%

""" 

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