  Consider the problem of efficiently evaluating isogenies $\phi: E \to E/H$ of
elliptic curves over a finite field $\mathbb{F}_q$, where the kernel $H =
\langle G\rangle$ is a cyclic group of odd (prime) order: given $E$, $G$, and a
point (or several points) $P$ on $E$, we want to compute $\phi(P)$. This
problem is at the heart of efficient implementations of group-action- and
isogeny-based post-quantum cryptosystems such as CSIDH. Algorithms based on
V{\'e}lu's formulae give an efficient solution to this problem when the kernel
generator $G$ is defined over $\mathbb{F}_q$. However, for general isogenies,
$G$ is only defined over some extension $\mathbb{F}_{q^k}$, even though
$\langle G\rangle$ as a whole (and thus $\phi$) is defined over the base field
$\mathbb{F}_q$; and the performance of V{\'e}lu-style algorithms degrades
rapidly as $k$ grows. In this article we revisit the isogeny-evaluation problem
with a special focus on the case where $1 \le k \le 12$. We improve
V{\'e}lu-style isogeny evaluation for many cases where $k = 1$ using special
addition chains, and combine this with the action of Galois to give greater
improvements when $k > 1$.
