\\ ---------------  GP code  ---------------------------------------
\\
\\ Time-stamp: <Mon, Jan 13, 2003 - 13:54:32 - apacetti>
\\
\\ Description: routine for computing the associated cubic to a quartic
\\
\\
\\ Original Author:     Fernando Rodriguez-Villegas 
\\                      villegas@math.utexas.edu
\\                      University of Texas at Austin
\\
\\
\\ With the assistance of:  Ariel Pacetti
\\			    apacetti@math.utexas.edu
\\			    University of Texas at Austin
\\
\\ Created:             April  2000
\\
\\-----------------------------------------------------------------

\\ Given a quartic f(x) will compute associated cubic g(x) so that 
\\ y^2=g(x) is the Jacobian of y^2=f(x) following Weil (Works II p. 111,
\\ Springer Verlag (1980).

jacobian(f)=
   {local(a,b,c,d,e);
   a=polcoeff(f,4);b=polcoeff(f,3);
   c=polcoeff(f,2);d=polcoeff(f,1);e=polcoeff(f,0);
   x^3+c*x^2+(b*d-4*a*e)*x-(4*a*c*e-b^2*e-a*d^2)
}

{helpjac_quart()=
print("-------------------------------------------------------------------");
print("      		Computational number Theory                        "); 
print("-------------------------------------------------------------------");
print("");
print("Description: routine for computing given a quartic f(x) the");
print(" associated cubic g(x) so that y^2=g(x) is the Jacobian of y^2=f(x)");
print("");
print("Original Author:     Fernando Rodriguez-Villegas");
print("                     villegas@math.utexas.edu");
print("                     University of Texas at Austin");
print("");
print("With the assistance of:  Ariel Pacetti");
print("                         apacetti@math.utexas.edu");
print("                         University of Texas at Austin");
print("-------------------------------------------------------------------");
print("");
print("Help functions");
print("");
print("jacobian");
print("");
}

addhelp(jacobian,"Given a quartic f(x) will compute associated cubic g(x) so that  y^2=g(x) is the Jacobian of y^2=f(x) following Weil (Works II p. 111, Springer Verlag (1980).");