sd22: greenberg -- mod 7 prob

700 days ago by wstein

a_2 is -2,-1,0,1,2

a_2 mod 7 is 0,1,2,5,6.
M = ModularSymbols(23,2,sign=1); M 
        
Modular Symbols space of dimension 3 for Gamma_0(23) of weight 2 with
sign 1 over Rational Field
Modular Symbols space of dimension 3 for Gamma_0(23) of weight 2 with sign 1 over Rational Field
def A(p): M = ModularForms(p,2) N = M.newforms('a'); N f = N[0] K = f.base_ring() return K.factor(7), f 
       
F, f = A(29); F 
        
(Fractional ideal (-2*a0 - 1)) * (Fractional ideal (-2*a0 - 3))
(Fractional ideal (-2*a0 - 1)) * (Fractional ideal (-2*a0 - 3))
P = F[0][0]; P 
        
Fractional ideal (-2*a0 - 1)
Fractional ideal (-2*a0 - 1)
P.residue_class_degree() 
        
1
1
f 
        
q + a0*q^2 - a0*q^3 + (-2*a0 - 1)*q^4 - q^5 + O(q^6)
q + a0*q^2 - a0*q^3 + (-2*a0 - 1)*q^4 - q^5 + O(q^6)
f[2].minpoly().roots(GF(7)) 
        
[(3, 1), (2, 1)]
[(3, 1), (2, 1)]

Take reduction where a_2 mod 7  is 3, and get that it doesn't come from elliptic curve.
float(2*sqrt(3)) 
        
3.4641016151377544
3.4641016151377544