Class group frequencies of real quadratic function fields: The degree 4 case

The distribution of ideal class groups of $\mathbb{F}_{q}(T,\sqrt {M(T)})$ is examined for degree-four monic polynomials $M \in \mathbb{F}_{q}[T]$ when $\mathbb{F}_{q}$ is a finite field of characteristic greater than 3 with $q \in [20000,100000]$ or $q \in [1020000,1100000]$ and $M$ is irreducible or has an irreducible cubic factor. Particular attention is paid to the distribution of the $p$-Sylow part of the class group, and these results agree with those predicted using the Cohen-Lenstra heuristics to within about 1 part in 10000. An alternative set of conjectures specific to the cases under investigation is in even sharper agreement.