by $\mu$ in ${\mathcal{O}}$ of reduced norm $\ell^{e_1}$.

Here we denote the element $\mu + N{\mathcal{O}}$ of ${\mathcal{O}}/N{\mathcal{O}}$ by $[\mu]$ to distinguish it from the conjugate $\bar{\mu}$ of $\mu $. The output $\beta = \gamma\mu$ is then an element of $I$ with reduced norm $N\ell^e$ where $e = e_0+e_1$. The element $\gamma$ can be constructed with the algorithm of Section \[sec:repns\_in\_orders\]. We solve for $[\mu]$ by linear algebra in Section \[sec:ideals:ellpow:isom\], showing that we can take $[\mu]$ in $(R/NR)^*[j] \subseteq ({\mathcal{O}}/N{\mathcal{O}})^*$. The core of the algorithm is the final specialized strong approximation algorithm of Section \[sec:ideals:ellpow:lift\], taking $[\mu]$ in $(R/NR)^*[j]$ and constructing the lifting $\mu$ of norm $\ell^e$. The whole algorithm for $p$-extremal orders is analyzed in Section \[sec:ideals:ellpow:results\].

As mentioned above, we finally remove the $p$-extremal condition in Section \[sec:ideals:gen\] by providing a reduction from the general case to the case of $p$-extremal orders, and we generalize our approach to compute ideal representatives of smooth or powersmooth norms in Section \[sec:ideals:psmooth\].

Effective strong approximation\[sec:ideals:strongapproximation\]
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Let $B := B_{p,\infty}$ be the quaternion algebra ramified at $p$ and $\infty$. Let ${\mathbb{A}}_{\mathbb{Q}}$ be the rational adèle ring, defined as the restricted product of ${\mathbb{Q}}_v$ with respect to ${\mathbb{Z}}_v$, let $\ell \ne 