//1. compute f2 in main theorem 3.1 over Rational Field,
//2. count the number of term in f2, 
//3. check f2 is NOT homogeneous polynomial,
//4. check f2 is irreducible polynomials


K:=Rationals();
INT:=Integers();

R<x,y,A,B,a0,a1,a2,a3,a4,a5>:=PolynomialRing(K,10,"grevlex");
S<X,Y,Z,AA,BB,aa0,aa1,aa2,aa3,aa4,aa5>:=PolynomialRing(K,11);
//compute polynomial f2 given in Main Theorem3.1-------------------- 
NonsingPolynomial:=function();
  PP:=Y^2*Z-(X^3+AA*X*Z^2+BB*Z^3);
  qq:=aa0*X^2 + aa1*X*Y + aa2*X*Z^2 + aa3*Y^2*Z + aa4*Y*Z + aa5*Z^2;
  //phi := hom<Sprime->Rprime | x,y,1,A,B>;
  phi := hom<S->R| x,y,1,A,B,a0,a1,a2,a3,a4,a5>;
  P:=phi(PP);
  q:=phi(qq);
  Res := Resultant(P,q,y);
  nonsing:=[R!0,R!0,R!0];
  
 Start_D:=Realtime();
  Disc := Discriminant(Res,x);
  FactDisc := Factorization(Disc);
  nonsing[1]:=4*A^3+27*B^2;
  nonsing[3]:=FactDisc[1][1];
  nonsing[2]:=Disc div (nonsing[3] ^ (FactDisc[1][2]));
 End_D:=Realtime();
  printf"Compute time Discriminant= %o\n",End_D - Start_D;

  return nonsing;
end function;

//-------------compute the number of terms in f2-------
TermCounter:=function(f)
  mono:=Monomials(f);
  return #mono;
end function;


F:=NonsingPolynomial();
f2:=F[2];
"----------------------f2-------------------------";
"f2=";
f2;
"------------------f2 end-------------------------";
N:=TermCounter(f2);
printf"f2 has %o terms \n",N;
"Is f2 homogeneous?";
IsHomogeneous(f2);
I:=ideal<R|f2>;
"Is f2 irreducible?";
IsPrime(I);