theorem is sufficient to solve the quaternion $\ell$-isogeny path problem. In particular, since $\beta$ is in $I$, the ideal $I\bar\beta/N$ is an isomorphic ideal of norm $\ell^e$.

Similarly, solving for $$\beta \equiv \lambda\alpha \bmod N{\mathcal{O}}$$ with $\lambda \in {\mathbb{Z}}$ coprime to $N$ such that we still have $I = {\mathcal{O}}(N,\beta)$, is also sufficient to solve the quaternion $\ell$-isogeny path problem. We will focus on this relaxed effective strong approximation theorem in the next subsections.

Isomorphism of ${\mathcal{O}}/N{\mathcal{O}}$-ideals {#sec:ideals:ellpow:isom}
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In this section, let $I$ be a left ${\mathcal{O}}$-ideal of prime norm $N \ne p$, and let $\gamma$ be an arbitrary element of ${\mathcal{O}}$ of norm $NM$, where $\gcd(N,M) = 1$. Since $N$ is large, we can assume that it does not divide the index $[{\mathcal{O}}:R+Rj]$, hence we have equalities of rings $${\mathcal{O}}/N{\mathcal{O}}= (R+Rj)/N(R + Rj) \cong {\mathbb{M}}_2({\mathbb{Z}}/N{\mathbb{Z}}).$$ We denote by $[\alpha]$ the class of an element $\alpha$ in ${\mathcal{O}}/N{\mathcal{O}}$ (as distinct from its conjugate $\bar{\alpha}$).

We note that ${\mathcal{O}}\gamma/N{\mathcal{O}}$ and $I/N{\mathcal{O}}$ are proper nonzero left ${\mathcal{O}}/N{\mathcal{O}}$-ideals. The following explicit classification of such ideals, in ${\mathbb{M}}_2({\mathbb{Z}}/N{\mathbb{Z}})$, will let us construct an explicit isomorphism between these ideals.

Let $N$ be a prime and $A = {\mathbb{M}}_2({\mathbb{Z}}/N{\mathbb{Z}})$. There exists a bijection $$S : 