2011-07-06-torsion7-finding 145 via torsion family -- Sage

K.<a> = NumberField(x^2 - x - 1)

def f(t): return EllipticCurve([-t^2+t+1,-t^3+t^2,-t^3+t^2,0,0])

f(2).torsion_order()

f(2)

Elliptic Curve defined by y^2 - x*y - 4*y = x^3 - 4*x^2 over Rational
Field

Elliptic Curve defined by y^2 - x*y - 4*y = x^3 - 4*x^2 over Rational Field

f(3).conductor()

B=3 for i in [-B..B]: for j in [-B..B]: t = i + j*a try: E = f(t) N = E.conductor() print i, j, N.norm() except ArithmeticError: pass

-3 -3 2908881
-3 -2 4906220
-3 -1 4248761
-3 0 459684
-3 1 221705
-3 2 1676
-3 3 159345
-2 -3 639
-2 -2 135476
-2 -1 221705
-2 0 86436
-2 1 905
-2 2 1676
-2 3 512721
-1 -3 145
-1 -2 1676
-1 -1 905
-1 0 676
-1 1 41
-1 2 20420
-1 3 508321
0 -3 159345
0 -2 1676
0 -1 41
Traceback (click to the left of this block for traceback)
...
ArithmeticError: Invariants [1, 0, 0, 0, 0] define a singular curve.

-3 -3 2908881
-3 -2 4906220
-3 -1 4248761
-3 0 459684
-3 1 221705
-3 2 1676
-3 3 159345
-2 -3 639
-2 -2 135476
-2 -1 221705
-2 0 86436
-2 1 905
-2 2 1676
-2 3 512721
-1 -3 145
-1 -2 1676
-1 -1 905
-1 0 676
-1 1 41
-1 2 20420
-1 3 508321
0 -3 159345
0 -2 1676
0 -1 41
Traceback (most recent call last):             print i, j, N.norm()
  File "", line 1, in <module>
    
  File "/tmp/tmpfRxVA0/___code___.py", line 4, in <module>
    exec compile(u'for i in (ellipsis_range(-B,Ellipsis,B)):\n    for j in (ellipsis_range(-B,Ellipsis,B)):\n         t = i + j*a\n         E = f(t)\n         N = E.conductor()\n         print i, j, N.norm()
  File "", line 4, in <module>
    
  File "/tmp/tmp8_vFup/___code___.py", line 4, in f
    return EllipticCurve([-t**_sage_const_2 +t+_sage_const_1 ,-t**_sage_const_3 +t**_sage_const_2 ,-t**_sage_const_3 +t**_sage_const_2 ,_sage_const_0 ,_sage_const_0 ])
  File "/sagenb/sage_install/sage-4.7/local/lib/python2.6/site-packages/sage/schemes/elliptic_curves/constructor.py", line 325, in EllipticCurve
    return ell_number_field.EllipticCurve_number_field(x, y)
  File "/sagenb/sage_install/sage-4.7/local/lib/python2.6/site-packages/sage/schemes/elliptic_curves/ell_number_field.py", line 162, in __init__
    EllipticCurve_field.__init__(self, [field(x) for x in ainvs])
  File "/sagenb/sage_install/sage-4.7/local/lib/python2.6/site-packages/sage/schemes/elliptic_curves/ell_generic.py", line 164, in __init__
    "Invariants %s define a singular curve."%ainvs
ArithmeticError: Invariants [1, 0, 0, 0, 0] define a singular curve.

E = f(-1 - 3*a); E

Elliptic Curve defined by y^2 + (-18*a-10)*x*y + (105*a+65)*y = x^3 +
(105*a+65)*x^2 over Number Field in a with defining polynomial x^2 - x -
1

Elliptic Curve defined by y^2 + (-18*a-10)*x*y + (105*a+65)*y = x^3 + (105*a+65)*x^2 over Number Field in a with defining polynomial x^2 - x - 1

def ap(E,p): return E.change_ring(p.residue_field()).trace_of_frobenius()

ap(E, K.primes_above(2)[0])

-2

-2

2011-07-06-torsion7-finding 145 via torsion family

322 days ago by WilliamStein