sd22 - finding half P

701 days ago by wstein

E = EllipticCurve('389a') P = E([0,0]) 
       
f = P.division_points(2, poly_only=True); f 
        
x^4 + 4*x^2 - 2*x + 3
x^4 + 4*x^2 - 2*x + 3
K.<a> = NumberField(f) 
       
F = E.change_ring(K); F 
        
Elliptic Curve defined by y^2 + y = x^3 + x^2 + (-2)*x over Number Field
in a with defining polynomial x^4 + 4*x^2 - 2*x + 3
Elliptic Curve defined by y^2 + y = x^3 + x^2 + (-2)*x over Number Field in a with defining polynomial x^4 + 4*x^2 - 2*x + 3
P2 = F(P); P2 
        
(0 : 0 : 1)
(0 : 0 : 1)
R = P2.division_points(2); R 
        
[(a : a^3 + 2*a - 1 : 1)]
[(a : a^3 + 2*a - 1 : 1)]
2 * R[0] 
        
(0 : 0 : 1)
(0 : 0 : 1)
-R[0] 
        
(a : -a^3 - 2*a : 1)
(a : -a^3 - 2*a : 1)
R.<y> = K[] M.<b> = K.extension(y^2 + y - (a^3+a^2-2*a)) 
        
Traceback (click to the left of this block for traceback)
...
ValueError: defining polynomial (y^2 + y - a^3 - a^2 + 2*a) must be
irreducible
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_27.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("Ui48eT4gPSBLW10KTS48Yj4gPSBLLmV4dGVuc2lvbih5XjIgKyB5IC0gKGFeMythXjItMiphKSk="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/private/var/folders/FE/FEo498bGEOeewp0B4AIIP++++TM/-Tmp-/tmpOhDdZ1/___code___.py", line 4, in <module>
    exec compile(u"M = K.extension(y**_sage_const_2  + y - (a**_sage_const_3 +a**_sage_const_2 -_sage_const_2 *a), names=('b',)); (b,) = M._first_ngens(1)" + '\n', '', 'single')
  File "", line 1, in <module>
    
  File "/Users/wstein/sage/build/sage/local/lib/python2.6/site-packages/sage/rings/number_field/number_field.py", line 3288, in extension
    return NumberField_relative(self, poly, str(name), check=check, embedding=embedding)
  File "/Users/wstein/sage/build/sage/local/lib/python2.6/site-packages/sage/rings/number_field/number_field_rel.py", line 300, in __init__
    raise ValueError, "defining polynomial (%s) must be irreducible"%polynomial
ValueError: defining polynomial (y^2 + y - a^3 - a^2 + 2*a) must be irreducible
E.defining_polynomial() 
        
-x^3 - x^2*z + y^2*z + 2*x*z^2 + y*z^2
-x^3 - x^2*z + y^2*z + 2*x*z^2 + y*z^2
E.defining_polynomial().subs(y=y,z=1,x=a) 
        
y^2 + y - a^3 - a^2 + 2*a
y^2 + y - a^3 - a^2 + 2*a
factor(y^2 + y - (a^3+a^2-2*a)) 
        
(y - a^3 - 2*a + 1) * (y + a^3 + 2*a)
(y - a^3 - 2*a + 1) * (y + a^3 + 2*a)
 
       
P.division_points(2, extend=True) 
        
Traceback (click to the left of this block for traceback)
...
TypeError: division_points() got an unexpected keyword argument 'extend'
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_31.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("UC5kaXZpc2lvbl9wb2ludHMoMiwgZXh0ZW5kPVRydWUp"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/private/var/folders/FE/FEo498bGEOeewp0B4AIIP++++TM/-Tmp-/tmpveYVG_/___code___.py", line 3, in <module>
    exec compile(u'P.division_points(_sage_const_2 , extend=True)
  File "", line 1, in <module>
    
TypeError: division_points() got an unexpected keyword argument 'extend'