//
//	code for Theorem 2.2 for N=40
//





C:=SmallModularCurve(40);		//gives a modular curve X_0(40)
SimplifiedModel(C);			//gives C=X_0(40) in the form y^2=f_40(x)





A:={};
S:=[0..3];
for m in S do
    for n in S do
        if ((m mod 3) ne 0) or ((n mod 3) ne 0) then					//because (m,n)=1
            f:=m^8+8*m^6*n^2-2*m^4*n^4+8*m^2*n^6+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 3};							//we put the residues modulo 3 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 3, for Ds^2 from (2)





A:={};
S:=[0..5];
for m in S do
    for n in S do
        if ((m mod 5) ne 0) or ((n mod 5) ne 0) then					//because (m,n)=1
            f:=m^8+8*m^6*n^2-2*m^4*n^4+8*m^2*n^6+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 5};							//we put the residues modulo 5 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 5, for Ds^2 from (2)