2011-07-07-reu_working with ben on tables

322 days ago by WilliamStein

var('x') K.<a> = NumberField(x^2 - x - 1) E = EllipticCurve([1,a+1,a,a,0]) 
       
import psage.ellcurve.lseries.sqrt5 as sqrt5w def aplist(E, B): # doesn't give 2, 3, 5 yet v = sqrt5w.TracesOfFrobenius(E, B) v._compute_split_traces() return v.aplist() 
       
timeit('v = aplist(E, 80)') 
        
625 loops, best of 3: 364 µs per loop
625 loops, best of 3: 364 µs per loop
V = [b[0] for b in v if b[2] is not None]; V 
        
[11, 11, 19, 19, 29, 29, 31, 41, 41, 59, 59, 61, 61, 71, 71, 79, 79, 89,
89]
[11, 11, 19, 19, 29, 29, 31, 41, 41, 59, 59, 61, 61, 71, 71, 79, 79, 89, 89]
pr = 1 i = 0 while pr < 2^63: pr *= int(ceil(2*sqrt(V[i]))) if pr >= 2^63: break i += 1 
       
V[:17] 
        
[11, 11, 19, 19, 29, 29, 31, 41, 41, 59, 59, 61, 61, 71, 71, 79, 79]
[11, 11, 19, 19, 29, 29, 31, 41, 41, 59, 59, 61, 61, 71, 71, 79, 79]
float(log(2000*3600*24*24,2)) 
        
31.949490477321433
31.949490477321433
 
       
L.<t> = K[] Et = EllipticCurve([-t^2+t+1,-t^3+t^2,-t^3+t^2,0,0]) ; Et 
        
Elliptic Curve defined by y^2 + (-t^2+t+1)*x*y + (-t^3+t^2)*y = x^3 +
(-t^3+t^2)*x^2 over Univariate Polynomial Ring in t over Number Field in
a with defining polynomial x^2 - x - 1
Elliptic Curve defined by y^2 + (-t^2+t+1)*x*y + (-t^3+t^2)*y = x^3 + (-t^3+t^2)*x^2 over Univariate Polynomial Ring in t over Number Field in a with defining polynomial x^2 - x - 1
Et.discriminant().factor() 
        
(t - 1)^7 * t^7 * (t^3 - 8*t^2 + 5*t + 1)
(t - 1)^7 * t^7 * (t^3 - 8*t^2 + 5*t + 1)
def G(t): return EllipticCurve([-t^2+t+1,-t^3+t^2,-t^3+t^2,0,0]) 
       
E = G(2*a-3); E 
        
Elliptic Curve defined by y^2 + (10*a-15)*x*y + (-42*a+68)*y = x^3 +
(-42*a+68)*x^2 over Number Field in a with defining polynomial x^2 - x -
1
Elliptic Curve defined by y^2 + (10*a-15)*x*y + (-42*a+68)*y = x^3 + (-42*a+68)*x^2 over Number Field in a with defining polynomial x^2 - x - 1
E.discriminant() 
        
283169694848*a - 458178190848
283169694848*a - 458178190848
t = 2*a-3; (t - 1)^7 * t^7 * (t^3 - 8*t^2 + 5*t + 1) 
        
283169694848*a - 458178190848
283169694848*a - 458178190848
E.conductor().factor() 
        
(Fractional ideal (2)) * (Fractional ideal (a - 21))
(Fractional ideal (2)) * (Fractional ideal (a - 21))
E.discriminant().factor() 
        
(-102334155*a + 165580141) * 2^7 * (a - 21)
(-102334155*a + 165580141) * 2^7 * (a - 21)
t = 2*a-3; factor(t^3 - 8*t^2 + 5*t + 1), factor((t - 1)^7), factor(t^7) 
        
((-5*a + 8) * (a - 21), (377*a - 610) * 2^7, 10946*a - 17711)
((-5*a + 8) * (a - 21), (377*a - 610) * 2^7, 10946*a - 17711)
N = -5*a+30; N 
        
-5*a + 30
-5*a + 30
N.norm().factor() 
        
5^2 * 29
5^2 * 29
factor(N) 
        
(-a + 6) * (2*a - 1)^2
(-a + 6) * (2*a - 1)^2
R.<t> = K[] (t^3 - 8*t^2 + 5*t + 1 - (2*a-1)^).roots() 
        
[]
[]
 
       
timeit('EllipticCurve([1,a+1,a,a,0]).conductor()') 
        
5 loops, best of 3: 41.5 ms per loop
5 loops, best of 3: 41.5 ms per loop
E 
        
Elliptic Curve defined by y^2 + x*y + a*y = x^3 + (a+1)*x^2 + a*x over
Number Field in a with defining polynomial x^2 - x - 1
Elliptic Curve defined by y^2 + x*y + a*y = x^3 + (a+1)*x^2 + a*x over Number Field in a with defining polynomial x^2 - x - 1
E.conductor().norm() 
        
31
31
E.quadratic_twist(6).rank() 
        
Traceback (click to the left of this block for traceback)
...
ValueError: There is insufficient data to determine the rank - 2-descent
gave lower bound 1 and upper bound 2
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_13.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("RS5xdWFkcmF0aWNfdHdpc3QoNikucmFuaygp"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/tmp/tmp7fGEx1/___code___.py", line 3, in <module>
    exec compile(u'E.quadratic_twist(_sage_const_6 ).rank()
  File "", line 1, in <module>
    
  File "/sagenb/sage_install/sage-4.7/local/lib/python2.6/site-packages/sage/schemes/elliptic_curves/ell_number_field.py", line 1712, in rank
    raise ValueError, 'There is insufficient data to determine the rank - 2-descent gave lower bound %s and upper bound %s' % (lower, upper)
ValueError: There is insufficient data to determine the rank - 2-descent gave lower bound 1 and upper bound 2
def primes_of_bounded_norm(B): v = sum([K.primes_above(p) for p in primes(B)],[]) v = [(p.norm(), p) for p in v if p.norm() <= B] v.sort() return [p[1] for p in v] 
       
v = primes_of_bounded_norm(100); v 
        
[Fractional ideal (2), Fractional ideal (2*a - 1), Fractional ideal (3),
Fractional ideal (3*a - 2), Fractional ideal (3*a - 1), Fractional ideal
(-4*a + 1), Fractional ideal (-4*a + 3), Fractional ideal (-a + 6),
Fractional ideal (a + 5), Fractional ideal (5*a - 2), Fractional ideal
(5*a - 3), Fractional ideal (a - 7), Fractional ideal (a + 6),
Fractional ideal (7), Fractional ideal (7*a - 2), Fractional ideal (7*a
- 5), Fractional ideal (7*a - 3), Fractional ideal (7*a - 4), Fractional
ideal (a - 9), Fractional ideal (a + 8), Fractional ideal (-8*a + 5),
Fractional ideal (-8*a + 3), Fractional ideal (a - 10), Fractional ideal
(a + 9)]
[Fractional ideal (2), Fractional ideal (2*a - 1), Fractional ideal (3), Fractional ideal (3*a - 2), Fractional ideal (3*a - 1), Fractional ideal (-4*a + 1), Fractional ideal (-4*a + 3), Fractional ideal (-a + 6), Fractional ideal (a + 5), Fractional ideal (5*a - 2), Fractional ideal (5*a - 3), Fractional ideal (a - 7), Fractional ideal (a + 6), Fractional ideal (7), Fractional ideal (7*a - 2), Fractional ideal (7*a - 5), Fractional ideal (7*a - 3), Fractional ideal (7*a - 4), Fractional ideal (a - 9), Fractional ideal (a + 8), Fractional ideal (-8*a + 5), Fractional ideal (-8*a + 3), Fractional ideal (a - 10), Fractional ideal (a + 9)]