  Orders on surfaces provide a rich source of examples of noncommutative
surfaces. Other than some existence results, very little is known about the
various moduli spaces that can be associated to them. Even fewer examples have
been explicitly computed. In this paper we compute the Picard and Hilbert
schemes of an order on the projective plane ramified on a union of two conics.
Our main result is that, upon carefully selecting the right Chern classes, the
Hilbert scheme is a ruled surface over a genus two curve. Furthermore, this
genus two curve is, in itself, the Picard scheme of the order.
