Let E be an elliptic curve of conductor N. Given a cyclic subgroup C
of order N in E, we construct a modular point P_C on E, called
self-point, as the image of (E,C) on X_0(N) under the modular
parametrisation X_0(N)->E. If N=p is prime, we prove that the point is
of infinite order in the Mordell-Weil group of E over the field of
definition of C. The study of these points in the PGL_2(Z_p)-tower
inside Q(E[p^infty]) continue earlier work of Harris.