  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 $K={\mathbb Q}$) 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 $E/{\mathbb Q}$ with large discriminant.
