  Let $(A,\mathfrak m)$ be an excellent two-dimensional normal local domain. In
this paper we study the elliptic and the strongly elliptic ideals of $A$ with
the aim to characterize elliptic and strongly elliptic singularities, according
to the definitions given by Wagreich and by Yau. In analogy with the rational
singularities, in the main result we characterize a strongly elliptic
singularity in terms of the normal Hilbert coefficients of the integrally
closed $\mathfrak m$-primary ideals of $A$. Unlike $p_g$-ideals, elliptic
ideals and strongly elliptic ideals are not necessarily normal and necessary
and sufficient conditions for being normal are given. In the last section we
discuss the existence (and the effective construction) of strongly elliptic
ideals in any two-dimensional normal local ring.
