Number of conjugacy classes in general affine group of fixed degree over a finite field is polynomial function of field size

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Statement

Suppose n is a natural number. Then, there exists a polynomial function f of degree n such that, for any prime power q, the number of conjugacy classes in the general affine group GA(n,q) (i.e., the general affine group of degree n over the finite field of size q) is f(q).

General observations

Below, we list some general observations about the polynomial in q giving number of conjugacy classes in GA(n,q).

Item Value
Degree of polynomial n
Leading coefficient of polynomial 1, i.e., it is always a monic polynomial
Factors of polynomial no common factors to all polynomials
Coefficients of polynomial The polynomial is an integer-valued polynomial, i.e., it sends integers to integers. Is it also true that all coefficients of the polynomial are integers? Seems so from first few examples

Particular cases

n (degree of general affine group, degree of polynomial) polynomial of q giving number of conjugacy classes in GA(n,q) More information
1 q element structure of general affine group of degree one over a finite field
2 q^2 + q - 1 element structure of general affine group of degree two over a finite field
3 q^3 + q^2 - 1 element structure of general affine group of degree three over a finite field
4 q^4 + q^3 + q^2 - q - 1 element structure of general affine group of degree four over a finite field
5  ? element structure of general affine group of degree five over a finite field
6  ? element structure of general affine group of degree six over a finite field
7  ?