nf=nfinit(y^3-1009);
P = primes(20);

pr = idealprimedec(nf,1009)[1]; a = y^11*(y+1)*101; b = y^5*(y+3)*19;
vector(#P, i, nfhilbert(nf, a, P[i]*b, pr))

pr = idealprimedec(nf,19)[1]; a = 19^11*(y+1)*101; b = 19^5*(y+3)*19;
vector(#P, i, nfhilbert(nf, a, (1+P[i]*y)*b, pr))

/* old regression cases: */
nf=nfinit(y^2+1); pr=idealprimedec(nf,2)[1];
nfhilbert(nf, [1,1]~, 3, pr)

nfhilbert(nf,[1,-2]~,[1,-2]~, pr)

nf=nfinit(y^2+3);
nfhilbert(nf,[3,0]~,[3,0]~,idealprimedec(nf,2)[1])


\\ #1147
K = nfinit(x^5-23);
p = idealprimedec(K, 2)[1];
nfhilbert(K,x,-x^2-5*x,p)

K = nfinit(x^8 + 2*x^7 + 3*x^6 + 3*x^4 + 3*x^2 + 2*x + 3);
p = idealprimedec(K, 2)[1];
{
for (j = 1,10,
  setrand(j);
  a = vectorv(8,i,random(7));
  b = vectorv(8,i,random(7));
  c = vectorv(8,i,random(7));
  d = nfeltmul(K,b,c);
  print([nfhilbert(K, a,b,p), nfhilbert(K, a,c,p), nfhilbert(K, a,d,p)])
)
}
