Let $p \geq 5$ be a prime. Let $Y(p)$ be the fine moduli space representing elliptic curves + basis of the $p$-torsion over $\mathbb{Q}_p$ and let $Y_0(p)$ be the fine moduli space representing elliptic curves + point of $p$-torsion over $\mathbb{Q}_p$. We know that $Y_0(p)$ has a model over $\mathbb{Z}_p$ whose special fiber is a union of two copies of $Y_0(1) := \mathbb{P}^1$ meeting transversaly at supersingular points.

My question : what is the analogous description of the special fiber for $Y(p)$ (for some suitable model over $\mathbb{Z}_p$) ? What is the explicit description of the natural map $Y(p) \rightarrow Y_0(p)$ at the level of special fibers ?

I guess it's in Katz--Mazur, but the purpose of my question is to get an answer as self-contained as possible. I don't (necessarily) ask for the idea of the proof.