reu-2011-06-22-checking for isogenies

337 days ago by WilliamStein

E = EllipticCurve([1,1,1,-8,6]) 
       
E.torsion_order() 
        
1
1
E.Np(5) 
        
6
6
E.isogeny_class() 
        
([Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 8*x + 6 over
Rational Field, Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 -
208083*x - 36621194 over Rational Field], [ 1 37]
[37  1])
([Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 8*x + 6 over Rational Field, Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 208083*x - 36621194 over Rational Field], [ 1 37]
[37  1])
def frob(p): R.<x> = ZZ[] t = E.ap(p) return x^2 - t*x + p 
       
frob(3).change_ring(GF(37)).factor() 
        
(x + 14) * (x + 24)
(x + 14) * (x + 24)
for p in primes(100): print p, frob(p).change_ring(GF(37)).factor() 
        
2 (x + 9) * (x + 29)
3 (x + 14) * (x + 24)
5 x^2 + 5
7 (x + 17) * (x + 20)
11 (x + 10) * (x + 27)
13 (x + 4) * (x + 31)
17 (x + 12) * (x + 23)
19 (x + 11) * (x + 32)
23 (x + 8) * (x + 26)
29 (x + 1) * (x + 29)
31 (x + 10) * (x + 29)
37 x * (x + 29)
41 (x + 1) * (x + 4)
43 (x + 1) * (x + 6)
47 (x + 8) * (x + 29)
53 (x + 20) * (x + 23)
59 (x + 20) * (x + 27)
61 (x + 18) * (x + 26)
67 (x + 14) * (x + 18)
71 (x + 3) * (x + 36)
73 (x + 14) * (x + 29)
79 (x + 13) * (x + 26)
83 (x + 17) * (x + 31)
89 (x + 11) * (x + 35)
97 (x + 6) * (x + 10)
2 (x + 9) * (x + 29)
3 (x + 14) * (x + 24)
5 x^2 + 5
7 (x + 17) * (x + 20)
11 (x + 10) * (x + 27)
13 (x + 4) * (x + 31)
17 (x + 12) * (x + 23)
19 (x + 11) * (x + 32)
23 (x + 8) * (x + 26)
29 (x + 1) * (x + 29)
31 (x + 10) * (x + 29)
37 x * (x + 29)
41 (x + 1) * (x + 4)
43 (x + 1) * (x + 6)
47 (x + 8) * (x + 29)
53 (x + 20) * (x + 23)
59 (x + 20) * (x + 27)
61 (x + 18) * (x + 26)
67 (x + 14) * (x + 18)
71 (x + 3) * (x + 36)
73 (x + 14) * (x + 29)
79 (x + 13) * (x + 26)
83 (x + 17) * (x + 31)
89 (x + 11) * (x + 35)
97 (x + 6) * (x + 10)
for p in primes(100): print p, frob(p).change_ring(GF(31)).factor() 
        
2 x^2 + x + 2
3 (x + 10) * (x + 22)
5 x^2 + 5
7 x^2 + 7
11 (x + 12) * (x + 19)
13 (x + 8) * (x + 21)
17 x^2 + 29*x + 17
19 x^2 + 6*x + 19
23 (x + 7) * (x + 21)
29 x^2 + 24*x + 29
31 x * (x + 2)
37 (x + 10) * (x + 13)
41 (x + 16) * (x + 20)
43 (x + 3) * (x + 4)
47 x^2 + 16
53 (x + 10) * (x + 27)
59 (x + 16) * (x + 25)
61 x^2 + 7*x + 30
67 (x + 10) * (x + 16)
71 x^2 + 2*x + 9
73 x^2 + 6*x + 11
79 x^2 + 2*x + 17
83 x^2 + 11*x + 21
89 (x + 19) * (x + 21)
97 x^2 + 16*x + 4
2 x^2 + x + 2
3 (x + 10) * (x + 22)
5 x^2 + 5
7 x^2 + 7
11 (x + 12) * (x + 19)
13 (x + 8) * (x + 21)
17 x^2 + 29*x + 17
19 x^2 + 6*x + 19
23 (x + 7) * (x + 21)
29 x^2 + 24*x + 29
31 x * (x + 2)
37 (x + 10) * (x + 13)
41 (x + 16) * (x + 20)
43 (x + 3) * (x + 4)
47 x^2 + 16
53 (x + 10) * (x + 27)
59 (x + 16) * (x + 25)
61 x^2 + 7*x + 30
67 (x + 10) * (x + 16)
71 x^2 + 2*x + 9
73 x^2 + 6*x + 11
79 x^2 + 2*x + 17
83 x^2 + 11*x + 21
89 (x + 19) * (x + 21)
97 x^2 + 16*x + 4