= \deg(\varphi)$ and conversely every left ${\mathcal{O}}$-ideal arises in this way (see Kohel [@Kohel1996 Section 5.3]). In particular, given any isogeny $\psi: E_0 \rightarrow E_1$ of degree $m$, the left ${\mathcal{O}}$-ideal $J = I \hat{\varphi}\psi/n$ is an equivalent ideal of norm $m$, where $\hat\psi$ is the dual of $\psi$.

The problem we address in this work is to solve the quaternion version of the supersingular $\ell$-isogeny path problem: given $E_0$, $E_1$ and a small prime $\ell$, find an $\ell$-power isogeny from $E_0$ to $E_1$. Under this equivalence of categories, the analogous problem is the determination of a $\ell$-power norm left ${\mathcal{O}}$-ideal in the class of a given left ${\mathcal{O}}$-ideal $I$. After introducing the necessary background on quaternion orders and ideals in Section \[sec:quaternions\] and addressing some preliminary algorithmic problems in Sections \[sec:prel\], we solve the $\ell$-power norm problem in Section \[sec:ideals:ellpow\]. Subject to reasonable heuristics on the probability of finding suitable primes, we obtain a probabilistic algorithm which solves this problem in expected polynomial time. The experimental runtime agrees with the most optimistic predictions for the distribution of primes. The algorithm gives a clear distinction between the efficiency of the $\ell$-isogeny problem in the equivalent category of quaternion ideals, whereas 