A unified method of construction of elliptic curves with high MordellWeil rank

By using the twist theory, we reduce the problem of con-structing elliptic curves of rank n (n ≥ 1) with generators to the problem of finding rational points on a certain variety Vn. By parametrizing all rational points on Vn (1 ≤ n ≤ 7), we get all elliptic curves of at least rank n (n ≤ 7). 1. Introduction. The purpose of this paper is to describe a unified method of construction of elliptic curves with given Mordell-Weil rank, and to show it is powerful enough to produce every known example in principle. In view of the fact that there is no general algorithm to give an elliptic curve with high rank