Classical Invariants and 2-descent on Elliptic Curves

The classical theory of invariants of binary quartics is applied to the problem of determining the group of rational points of an elliptic curve defined over a field K by 2-descent. The results lead to some simplifications to the method first presented in Birch and Swinnerton-Dyer (1963), and can be applied to give a more efficient algorithm for determining Mordell–Weil groups over Q, as well as being more readily extended to other number fields. In this paper we mainly restrict ourselves to general theory, valid over arbitrary fields of characteristic neither 2 nor 3.