The de Rham cohomology of the Suzuki curves

For a natural number $m$, let $mathcal{S}_m/mathbb{F}_2$ be the $m$th Suzuki curve. We study the mod $2$ Dieudonn{e} module of $mathcal{S}_m$, which gives the equivalent information as the Ekedahl-Oort type or the structure of the $2$-torsion group scheme of its Jacobian. We accomplish this by studying the de Rham cohomology of $mathcal{S}_m$. For all $m$, we determine the structure of the de Rham cohomology as a $2$-modular representation of the $m$th Suzuki group and the structure of a submodule of the mod $2$ Dieudonn{e} module. For $m=1$ and $2$, we determine the complete structure of the mod $2$ Dieudonn{e} module.