Re: How to calculate the conductor of an abelian extension such as Q[x]/(x^3- 19x -19)

[Kurt Foster <drsardonicus@earthlink.net>]

On Nov 10, 2024, at 10:08 PM, David Bernier wrote:

> I'm interested in cubic extensions of Q that are abelian, in  
> connection with a probable prime test. I have a list of cubic  
> polynomials f_1, ... f_22 and I want to find the first f_i such that  
> f_i is irreducible over F_p, where p can be assumed prime. For a  
> given f_i, I noticed a periodicity in p of the irreducibility  
> character of f_i over F_p (ref. Mathematics Stack Exchange at the  
> link: https://math.stackexchange.com/questions/4995484/irreducibility-of-cubic-polynomials-over-finite-fields-f-p 
>  ). User leoli1 mentioned as relevant the conductor N of the  
> splitting field of f_i. I have f_8 = X^3 - 19X - 19 with  
> discriminant 133^2. How could I calculate the conductor of Q[x]/ 
> (f_8) in PARI/gp?

Karim and Bill have shown how to find the conductors.  The subgroup H  
is more specific. If the conductor has more than one prime factor,  
there will be more than one abelian cubic field with that conductor.   
As Bill pointed out, the polynomial discriminants can have extraneous  
factors.

Note that the answer to your original question could be "none."

In order to answer your initial question, let v = [f_1, f_2, ...  
f_22], and suppose p is the given prime.  Then you could do something  
like this:

for(i=1,22,M=factormod(v[i],p);if(M[1,2]==3,print(i"  
"v[i]);break);if(i==22,print("None"))