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.
Categories: math.NT