greenberg-2010-11-08-3tor

452 days ago by wstein

Question: Suppose that E is an elliptic curve over \QQ with a rational 3-torsion point.  Consider the field K=\QQ(E[3]), which is a cubic extension of \QQ(\zeta_3).      In fact, I can show that K = \QQ(\zeta_3)(\sqrt[3]{d}) for some integer d\in\ZZ.   Can you use Sage to compute d?
 
       
K3 = QuadraticField(-3) def three_d(E): for f, _ in E.division_polynomial(3).factor(): if f.degree() == 3: # found a factor of degree 3 g = sage.schemes.elliptic_curves.heegner.make_monic(f)[0] K = NumberField(g, 'a').composite_fields(K3)[0] assert K.degree() == 6, 'bug in code' # Figure out which d is such that x^3-d has a root in K. # Then Q(d^(1/3)) is a subfield of K, so its Galois closure # must be K, so it *must* be the right field. R.<x> = K[] P = E.conductor().prime_divisors() for e in cartesian_product_iterator([[0,1,2]]*len(P)): d = prod(P[i]^e[i] for i in range(len(P))) if d > 1 and len((x^3 - d).roots()) > 0: return d d *= -1 if d < -1 and len((x^3 - d).roots()) > 0: return d # no factor of degree 3, so mod-3 representation has to be split, so d=1 works. return 1 
       
for E in cremona_curves([1..100]): if E.torsion_order()==3: print E.cremona_label(), three_d(E).factor(), ", ", E.discriminant().factor() 
        
19a1 1 ,  -1 * 19^3
19a3 19 ,  -1 * 19
26a1 1 ,  -1 * 2^3 * 13^3
26a3 2 * 13 ,  -1 * 2 * 13
27a1 1 ,  -1 * 3^9
27a3 1 ,  -1 * 3^3
27a4 3 ,  -1 * 3^5
35a1 1 ,  -1 * 5^3 * 7^3
35a3 5 * 7 ,  -1 * 5 * 7
37b1 1 ,  37^3
37b3 37 ,  37
38a1 1 ,  -1 * 2^9 * 19^3
38a3 19 ,  -1 * 2^3 * 19
44a1 2 * 11^2 ,  -1 * 2^8 * 11
50a1 2 * 5 ,  -1 * 2 * 5^4
50a3 2 * 5 ,  -1 * 2^5 * 5^8
51a1 17 ,  -1 * 3^3 * 17
54a1 1 ,  -1 * 2^3 * 3^9
54a3 2 ,  -1 * 2 * 3^3
54b1 1 ,  -1 * 2^3 * 3^3
77b1 1 ,  -1 * 7^6 * 11^3
77b3 7 * 11^2 ,  -1 * 7^2 * 11
91b1 7 * 13 ,  -1 * 7 * 13
91b2 1 ,  -1 * 7^3 * 13^3
92a1 2 * 23 ,  -1 * 2^4 * 23
19a1 1 ,  -1 * 19^3
19a3 19 ,  -1 * 19
26a1 1 ,  -1 * 2^3 * 13^3
26a3 2 * 13 ,  -1 * 2 * 13
27a1 1 ,  -1 * 3^9
27a3 1 ,  -1 * 3^3
27a4 3 ,  -1 * 3^5
35a1 1 ,  -1 * 5^3 * 7^3
35a3 5 * 7 ,  -1 * 5 * 7
37b1 1 ,  37^3
37b3 37 ,  37
38a1 1 ,  -1 * 2^9 * 19^3
38a3 19 ,  -1 * 2^3 * 19
44a1 2 * 11^2 ,  -1 * 2^8 * 11
50a1 2 * 5 ,  -1 * 2 * 5^4
50a3 2 * 5 ,  -1 * 2^5 * 5^8
51a1 17 ,  -1 * 3^3 * 17
54a1 1 ,  -1 * 2^3 * 3^9
54a3 2 ,  -1 * 2 * 3^3
54b1 1 ,  -1 * 2^3 * 3^3
77b1 1 ,  -1 * 7^6 * 11^3
77b3 7 * 11^2 ,  -1 * 7^2 * 11
91b1 7 * 13 ,  -1 * 7 * 13
91b2 1 ,  -1 * 7^3 * 13^3
92a1 2 * 23 ,  -1 * 2^4 * 23