  We classify two dimensional neighborhoods of an elliptic curve C with torsion
normal bundle, up to formal equivalence. The proof makes use of the existence
of a pair (indeed a pencil) of formal foliations having C as a common leaf, and
the fact that neighborhoods are completely determined by the holonomy of such a
pair. We also discussanalytic equivalence and show, for each formal model, that
the corresponding moduli space is infinite dimensional.
