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 deened over a eld K by 2-descent. The results lead to some simpliications to the method rst presented in (Birch and Swinnerton-Dyer, 1963), and can be applied to give a more eecient algorithm for determining Mordell-Weil groups over Q, as well as being more readily extended to other number elds. In this paper we mainly restrict to general theory, valid over arbitrary elds of characteristic neither 2 nor 3.