Low-degree test polynomials with different signatures

[Kurt Foster <drsardonicus@earthlink.net>]

The usual lists of "test polynomials" for transitive groups of given  
degree n (for example http://world.std.com/~jmccarro/math/GaloisGroups/GaloisGroupPolynomials.html 
 for degrees up to 9) usually give only one polynomial of degree n  
per group.  However, there is usually more than one possibility for  
the signature.  I was unable to find a list giving polynomials with  
each possible signature per group.

I'm aware of the GALPOL database of defining polynomials for normal  
extensions of Q with Galois groups G  of order up to 110.  For those G  
which are weakly supersolvable, of course, I could probably use Pari's  
galoissubfields() function to get some examples with different  
signatures in some cases.

The particular case that got my attention was the degree-6 group 6T14  
= PGL(2,5) of order 120, just beyond the range of the GALPOL  
database.  It's not hard to show that a degree-6 polynomial in Q[x]  
with this Galois group can *not* have signature [4,1], but the test  
polynomial in the above-mentioned list has signature [0,3].  I assume  
signature [6,0] is possible, and have been too lazy to consider  
whether signature [2,2] is possible -- yet.

Of course for groups G of order n whose prime-order subgroups are all  
normal, irreducible polynomials in Q[x] with Galois group G must have  
degree n, so the zeroes must be either all real or all complex.

So I was wondering: Are there tables of all possible signatures for  
irreducible polynomials of degree n having a given transitive group G  
of degree n as Galois group for n up to 8 or 10 or something?  And if  
so, are there corresponding test polynomials for each case?

If there are no such tables, are there relatively simpleminded ways of  
constructing examples?