An elliptic curve is an object that has both the analytic structure of a Riemann Surface, and the algebraic structure of a group. Under this group structure, we can consider the cyclic subgroup generated by an algebraic point on the curve, and ask whether this subgroup is dense in the complex points on the curve, under the usual topology on the analytic structure.  We give conditions on the point in question for its multiples to be dense in the complex points on the curve. We discuss transcendence results for the Weierstrass [Weierstrass p] function, analogous to results of the same nature for the regular exponential function