//
//	code for Theorem 2.13, Table 7
//





function check(p,f);                                                  		  // this function returns the set of all possible residues of f mod p if p is not 2, and mod 4 if p is 2
A:={};                                                                   
S:=[0..p];
for m in S do
    for n in S do
        if ((m mod p) ne 0) or ((n mod p) ne 0) then 			         // because (m,n)=1
              g:=Integers()!Evaluate(f, [m, 0, n]);
              if p eq 2 then
                  A:=A join {-g mod 4};                                          // we put the residues modulo 4 in the set A
              else 
                  A:=A join {-g mod p};         	                      	 // we put the residues modulo p in the set A
              end if;
        end if;
    end for;
end for;
return A;
end function;


function Unramified(N);                                              	         // this function returns the set of primes between 1 and 100 that are unramified in all quadratic fields generated by points on X0(22)
f:=DefiningPolynomial(SimplifiedModel(SmallModularCurve(N)));
A:={i:i in PrimesUpTo(100)};
for p in PrimesUpTo(100) do
    B:=check(p,f);
    if p eq 2 then 
        for k in B do
            if B ne {1} then
                A:=A diff {2};					            	// if (k mod 4) is not 1, then 2 may ramify
            end if;					            	
        end for;
    else
        if 0 in B then
            A:=A diff {p};					            	// if 0 is in B, then p may ramify
        end if;
    end if;
end for;
return A;
end function;


Table4:=[*[*22, {3, 5, 23, 31, 37, 59, 67, 71, 89, 97}*],              						// [*N, {primes that are unramified in all quadratic fields generated by points on X0(N)}*], according to Theorem 2.8
          [*23, {2, 3, 13, 29, 31, 41, 47, 71, 73}*],
          [*26, {3, 5, 7, 11, 17, 19, 31, 37, 41, 43, 47, 59, 67, 71, 73, 83, 89, 97}*],
          [*28, {3, 5, 13, 17, 19, 31, 41, 47, 59, 61, 73, 83, 89, 97}*],
          [*29, {3, 5, 11, 13, 17, 19, 31, 37, 41, 43, 47, 53, 61, 73, 79, 89, 97}*],        			// we exclude p=2 here, it was proved in Theorem 2.1.a)
          [*30, {3, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97}*],
          [*31, {2, 5, 7, 19, 41, 59, 71, 97}*],
          [*33, {2, 7, 13, 17, 19, 29, 41, 43, 61, 73, 79, 83}*],
          [*35, {2, 3, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97}*],
          [*39, {2, 5, 7, 11, 13, 19, 31, 37, 41, 47, 59, 61, 67, 71, 73, 79, 83, 89, 97}*],
          [*40, {3, 5, 7, 11, 13, 17, 19, 23, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 97}*], 	// we exclude p=2 here, it was proved in Theorem 2.1.a)
          [*41, {3, 5, 7, 11, 13, 17, 19, 29, 37, 47, 53, 61, 67, 71, 73, 79, 89, 97}*],
          [*46, {3, 13, 29, 31, 41, 47, 71, 73}*],           							// we exclude p=2 here, because it was proved in Theorem 2.1.a)
          [*47, {2, 3, 7, 17, 37, 53, 59, 61, 71, 79, 89, 97}*],
          [*48, {3, 5, 7, 11, 17, 19, 23, 29, 31, 41, 43, 47, 53, 59, 67, 71, 79, 83, 89}*],		        // we exclude p=2 here, because it was proved in Theorem 2.1.a)
          [*50, {3, 7, 11, 13, 17, 19, 23, 37, 41, 43, 47, 53, 67, 73, 83, 89, 97}*],
          [*59, {3, 5, 7, 19, 29, 41, 53, 79}*],
          [*71, {2, 3, 5, 19, 29, 37, 43, 73, 79, 83, 89}*]*];	

			


A:={}; B:={};
for i in [1..#Table4] do
    N:=Table4[i][1];							
    B:=Table4[i][2];                                                   // the set of primes that are unramified in all quadratic fields generated by points on X0(N), according to Theorem 2.8
    A:=Unramified(N);    					       // the set of primes that are unramified in all quadratic fields generated by points on X0(N), according to the function Unramified()
    assert A eq B;
end for;