On the topology of elliptic curves

The paper is concerned with studying the everywhere density of the set of rational points on an elliptic curve. V. N. Ushakov's conjecture, which is related to the topology of elliptic curves, is investigated. Elliptic curves and Weierstrass functions with real invariants are considered. A criterion of the everywhere density of a set of points on a curve is proved. The rank of an elliptic curve is calculated numerically.