(16:46) gp > f=x^2 - x + 6;  \\ quadratic field of disc -23
(16:47) gp > F=bnfinit(f);   \\ a bnf has more data than what you get
                             \\ from nfinit, in particular info about the class group
                             \\ and the unit group 
(16:47) gp > F.clgp
%3 = [3, [3], [[2, 1; 0, 1]]]  \\ the class group of F is Z/3Z; see
                               \\ docs for more info 
(16:47) gp > pl=idealprimedec(F,2);  \\ primes over 2
(16:47) gp > #pl  \\ how many
%5 = 2            \\ 2, so split
(16:47) gp > p=pl[1];  \\ pick one
(16:47) gp > idealnorm(F,p)  \\ ideal norm
%7 = 2    \\ as expected
(16:48) gp > bnfisprincipal(F,p)  \\ is this ideal principal?
%8 = [[1]~, [1, 0]~]  \\ this means no, it's not.  in fact
                      \\ the [1]~ means it's a generator of the class group
(16:48) gp > pl=idealprimedec(F,3);  \\ primes over 3
(16:48) gp > q=pl[1]   \\ one of them
%10 = [3, [-1, 1]~, 1, 1, [0, -6; 1, 1]]  \\ gp's data for an ideal
(16:48) gp > idealnorm(F,q)  \\ ideal norm
%11 = 3  \\ again what we expect
(16:49) gp > pq = idealmul(F,p,q) \\ product of ideals p and q
%12 = 
[6 5]

[0 1]

(16:49) gp > idealnorm(F,pq)
%13 = 6  \\ this ideal has norm 6
(16:49) gp > bnfisprincipal(F,pq)  \\ is it principal?
%14 = [[0]~, [-1, 1]~]  \\ yes, the [0]~ means trivial class