Higher descents on an elliptic curve with a rational 2-torsion point

Let E be an elliptic curve over a number field K. Descent calculations on E can be used to find upper bounds for the rank of the Mordell-Weil group, and to compute covering curves that assist in the search for generators of this group. The general method of 4-descent, developed in the PhD theses of Siksek, Womack and Stamminger, has been implemented in Magma (when K = Q) and works well for elliptic curves with sufficiently small discriminant. By extending work of Bremner and Cassels, we describe the improvements that can be made when E has a rational 2-torsion point. In particular, when E has full rational 2-torsion, we describe a method for 8-descent that is practical for elliptic curves E/Q with large discriminant.