//
//	code for Theorem 2.2 for N=33
//





C:=SmallModularCurve(33);		//gives a modular curve X_0(33)
SimplifiedModel(C);			//gives C=X_0(33) in the form y^2=f_33(x)





A:={};
S:=[0..8];
for m in S do
    for n in S do
        if ((m mod 2) ne 0) or ((n mod 2) ne 0) then							//because (m,n)=1
            f:=m^8+10*m^6*n^2-8*m^5*n^3+47*m^4*n^4-40*m^3*n^5+82*m^2*n^6-44*m*n^7+33*n^8;		//this is n^8d=Ds^2 from (2), obtained by putting x=m/n in SimplifiedModel(C) from above and multiplying with the denominator n^8
            A:=A join {f mod 8};									//we put the residues modulo 8 in the set A
        end if;
    end for;
end for;
A;													//in the end the set A will contain all possible values of Ds^2 modulo 8, for Ds^2 from (2)