  For any positive integers $n\geq 3, r\geq 1$ we present formulae for the
number 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 algebraic
curves over finite fields, a technique which arose from Fourier-analysing the
known formulae for the $\mathbb{F}_2$ base field cases, reverse-engineering an
economical new proof and then extending it. This approach gives rise to fibre
products of supersingular curves and makes explicit why the formulae have
period $24$ in $n$.
