My question is related to this:
What is the relation between polynomial factorisation and field size?
Assume all operations and polynomials are defined and done on $\mathbb{F}_q$, where $q$ is a large prime number of size $z$ bits. As stated in the above post and here, the complexity of factorizing a polynomial of degree $n$ is $O(n^c log 2^z)$ , where $1< c \leq 2$.
Consider the case where we have two polynomials of the same degree one is defined on a field of size $112$-bit and the other one on a $40$-bit field. Then, to calculate the runtime difference between the two polynomials we can have:
$$\frac{\log(2^{112})}{\log(2^{40})}= 2.8.$$
But, when I run NTL library (so I have written the code in C++) to factorize two polynomials specified above the ratio would be about: $2.19$. That means the implementation is faster than the asymptotic analysis.
Question: Is my analysis above correct? If yes, why the running time in practice is faster than the asymptotic analysis?