  In this paper we consider smooth affine elliptic plane curves having one
place at infinity. We identify them with elliptic projective plane curves
having only one cusp as their singular points and meeting with the line at
infinity only at the cusp. We characterize such curves by the self-intersection
number of the strict transform of them via the minimal embedded resolution of
their cusp. Furthermore, we prove that the self-intersection number of them is
the maximum value among those of all the elliptic plane curves having only one
cusp.
