It is given by the Colmez conjecture that if $A$ has a CM-type $(k, \phi)$ and $\text{End}(A) = \mathcal{O_k}$ $$\displaystyle h_{\text{fal}}(A) = \sum_{\text{irr} \hspace{1 mm}\rho: G(k^{\text{nor}}/\mathbb{Q}) \to GL_n(\mathbb{C})} C_{\rho,\phi}\left (\dfrac{L'(1,p)}{L(1,p)} +\log fp\right)$$ where we are taking a sum of irreducible representations $\rho$ from the Galois group of the normal closure of $k$ over $\mathbb{Q}$ to some $GL_n(\mathbb{C}).$

My question is is it necessary for $\text{End}(A) = \mathcal{O_k}$? Can you get a formula in a different way or must you have the endomorphisms of $A$ be the full ring of integers?

Note that $\log fp$ is the artin conductor of $\rho$.