\\ ---------------  GP code  ---------------------------------------
\\
\\ Time-stamp: <Fri, Dec 20, 2002 - 12:20:46 - apacetti>
\\ Description: Compute list of primes where a given conic over \Q
\\ does not have points locally
\\
\\ File: /home/villegas/.public_html/cnt/conic.gp
\\
\\ Original Author:     Fernando Rodriguez-Villegas 
\\                      villegas@math.utexas.edu
\\                      University of Texas at Austin
\\
\\ Created:             Tue Oct  3 2000
\\
\\-----------------------------------------------------------------


\\----------------------------------------------------------------------
\\ Given a non-degenerate 3 by 3 symmetric matrix with rational
\\ entries compute the list of primes for which the corresponding
\\ conic does not have a rational point locally (including 0 as the
\\ infinite prime). The length of the answer should be even.

hilbertlist(m)=
        {local(u,a,b,p,pl,fdis);
	if(matdet(m),
        m=qfgaussred(m);u=[];
        a=-m[2,2]/m[1,1];
        b=-m[3,3]/m[1,1];
	pl=[0,2];
	fdis=factor(abs(a*b)/gcd(a,b));
	for(k=1,length(fdis~),
		if(fdis[k,2]%2&&fdis[k,1]>2,pl=concat(pl,fdis[k,1])));
	for(k=1,length(pl),
		if(hilbert(a,b,pl[k])==-1,u=concat(u,pl[k]),));
       u,
       error("matrix is singular"))
}

addhelp(kilbertlist,"Given a non-degenerate 3 by 3 symmetric matrix with rational entries compute the list of primes for which the corresponding conic does not have a rational point locally (including 0 as the infinite prime). The length of the answer should be even.")