  Elliptic curves have a well-known and explicit theory for the construction
and application of endomorphisms, which can be applied to improve performance
in scalar multiplication. Recent work has extended these techniques to
hyperelliptic Jacobians, but one obstruction is the lack of explicit models of
curves together with an efficiently computable endomorphism. In the case of
hyperelliptic curves there are limited examples, most methods focusing on
special CM curves or curves defined over a small field. In this article we
describe three infinite families of curves which admit an efficiently
computable endomorphism, and give algorithms for their efficient application.
