The de Rham cohomology of the Suzuki curves

For a natural number $m$, let ${\mathcal{S}}_m\mathrm{/}{\mathbb{F}}_2$ 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.