Let $d<0$ be a discriminant and fix an embedding $\mathbb{Q}(\sqrt{d})\to \mathbb{C}$. Let $\tau\in \mathbb{Q}(\sqrt{d})\cap \mathcal{H}$, then $j(\tau)$ is an algebraic number. Moreover, write $R:=\operatorname{End}(\Lambda_\tau)$, then $R$ is isomorphic to an order in $\mathbb{Q}(\sqrt{d})$ and if $R$ is the maximal order, then $j(\tau)$ is an algebraic integer.
Wikipedia, seems to suggest that $j(\tau)$ is an algebraic integer even when $R$ is non-maximal. However, it refers to Silverman, where this is proven only for $R$ maximal.
Questions: For what non-maximal orders $\operatorname{End}(\Lambda_\tau)$ is $j(\tau)$ an algebraic integer? Do you have examples of $\tau$ with $j(\tau)$ not an algebraic integer?