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

The projective characterization of elliptic plane curves which have one place at infinity