Division by 2 of rational points on elliptic curves

The well-known divisibility by 2 condition for rational points on elliptic curves with rational 2-torsion is reproved in a simple way. Next, the explicit formulas for division by 2ⁿ obtained in §2 are used to construct versal families of elliptic curves that contain points of orders 4, 5, 6, and 8. These families are further employed to describe explicitly elliptic curves over certain finite fields F[double-struck]_q with a prescribed (small) group E(F[double-struck]_q). The last two sections are devoted to the cases of 3- and 5-torsion.