  In this paper we consider elliptic divisibility sequences generated by a
point on an elliptic curve over $\mathbb{Q}$ with $j$-invariant $1728$ given by
an integral short Weierstrass equation. For several different such elliptic
divisibility sequences, we determine explicitly a finite set of primes such
that for all primes $l$ outside this set, the elliptic divisibility sequence
contains no $l$-th powers. Our approach uses, amongst other ingredients, the
modular method for $\mathbb{Q}$-curves. The corresponding explicit computations
fit into a general computational framework for $\mathbb{Q}$-curves, for which
they provide illustrative examples.
