2011-02-04-J0(389)

476 days ago by wstein

M = ModularSymbols(389,sign=1) S = M.cuspidal_submodule() D = S.decomposition() e = M([0,oo]) 
       
for A in D: e_imag = A.rational_period_mapping()(e) w = A.atkin_lehner_operator(389).matrix()[0,0] print A.dimension(), w, e 
        
1 -1 (0)
2 1 (0, 0)
3 1 (0, 0, 0)
6 1 (0, 0, 0, 0, 0, 0)
20 -1 (-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
1 -1 (0)
2 1 (0, 0)
3 1 (0, 0, 0)
6 1 (0, 0, 0, 0, 0, 0)
20 -1 (-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
EllipticCurve('389a').rank() 
        
2
2
G.<chi> = DirichletGroup(4) 
       
chi(-389) 
        
-1
-1
EllipticCurve('389a').quadratic_twist(-1).rank() 
        
1
1
389*16 
        
6224
6224

Rank over \QQ:  rank 2 curve + 2,3,6 rank 1 factors
2 + 2 + 3 + 6 
        
13
13

Rank of quadratic twist by \chi is (very probably):  rank 1 curve + rank 20 twist.

This could be proved by explicitly computing L(f,\chi,s) for each f, which is possible (but I don't have time now).

 
1 + 20 
        
21
21

So rank over \QQ(i)  is probably 34.