Elliptic curves and explicit enumeration of irreducible polynomials with two coefficients prescribed

Let Fq be a finite field of characteristic p=2,3. We give the number of irreducible polynomials xm+am−1xm−1+ +a0 Fq[x] with am−1 and am−3 prescribed for any given m if p=2, and with am−1 and a1 prescribed for m=1,…,10 if p=2,3. In the latter case an enumeration formula, applicable also if m>10, is given, but it is explicit only up to the evaluation of certain Kronecker class numbers.