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For any positive integers $n\geq 3, r\geq 1$ we present formulae for thenumber of irreducible polynomials of degree $n$ over the finite field$\mathbb{F}_{2^r}$ where the coefficients of $x^{n-1}$, $x^{n-2}$ and $x^{n-3}$are zero. Our proofs involve counting the number of points on certain algebraiccurves over finite fields, a technique which arose from Fourier-analysing theknown formulae for the $\mathbb{F}_2$ base field cases, reverse-engineering aneconomical new proof and then extending it. This approach gives rise to fibreproducts of supersingular curves and makes explicit why the formulae haveperiod $24$ in $n$.

Fibre Products of Supersingular Curves and the Enumeration of Irreducible Polynomials with Prescribed Coefficients