Let $p$ be a prime and $K$ a quadratic imaginary field in which $p$ splits. Let $\mathcal{O}$ be an order in $K$ and $A$ an elliptic curve with CM by $\mathcal{O}$. Then $A$ can be defined over the ring class field $H$ of the order $\mathcal{O}$. Assume there exists a prime $\mathfrak{p}$ of $H$ above $p$ at which $A$ has good reduction. Then this reduction is in fact ordinary, and thus $A$ has a distinguished subgroup $\Lambda$ of order $p$, the so-called canonical subgroup (see Chapter 3 of Katz in LNM350).

Can one give a different description of the isogeny $A\rightarrow A/\Lambda$? And what is the relation between ${\rm End}(A)$ and ${\rm End}(A/\Lambda)$?

Comments:

I am thinking in an answer generalizing the following: suppose $A$ defined over $K$ of class number one, and $p=PQ$ with $P\neq Q$. Then $\Lambda=A[P]$, so $A\rightarrow A/\Lambda$ is induced by the complex multiplication of $P$ on $A$, and ${\rm End}(A/\Lambda)={\rm End}(A)$.

I pressume the answer in general depends on whether the conductor of $\mathcal{O}$ is divisible by $p$ or not.