//
//	code for Theorem 2.2 for N=30
//





C:=SmallModularCurve(30);		//gives a modular curve X_0(30)
SimplifiedModel(C);			//gives C=X_0(30) in the form y^2=f_30(x)





A:={};
S:=[0..128];
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+14*m^7*n+79*m^6*n^2+242*m^5*n^3+441*m^4*n^4+484*m^3*n^5+316*m^2*n^6+112*m*n^7+16*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 128};											//we put the residues modulo 128 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 128, for Ds^2 from (2)





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+14*m^7*n+79*m^6*n^2+242*m^5*n^3+441*m^4*n^4+484*m^3*n^5+316*m^2*n^6+112*m*n^7+16*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..25];
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+14*m^7*n+79*m^6*n^2+242*m^5*n^3+441*m^4*n^4+484*m^3*n^5+316*m^2*n^6+112*m*n^7+16*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 25};											//we put the residues modulo 25 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 25, for Ds^2 from (2)