Let $\pi:\mathcal{E}\rightarrow U$ be a family of elliptic curves with level $n$ structure (in the sense of Deligne-Rapoport) where $U\subseteq \mathbf{P}^1(\mathbf{C})$ is some (non-empty) Zariski open set. Let us take $\tilde{\mathcal{E}}$
the minimal proper regular model of $\mathcal{E}$ over $\mathbf{C}((t))$ where $t$ is (an appropriate choice) a local parameter of $\mathbf{P}^1(\mathbf{C})$. Embedd $\tilde{\mathcal{E}}$ in 
$\mathbf{P}^m(\mathbf{C})$ (for some appropriate $m$) and assume that the defining equations of the embedding are defined over $\mathbf{C}[[t]]$. Now one may reduce the scheme
$\tilde{\mathcal{E}}$ modulo $t$ to obtain a scheme
$E'$ over $Spec(\mathbf{C})$. Assume that $E'$ is **not smooth**.

**Q1** Is it "possible" (for us humans) to write down explicit equations for such an embedding (at least for small values of $n$)?

**Q2** How does one prove (algebraically and/or analytically) that $E'$ is isomorphic (as algebraic variety) to 
$$
\mathbf{P}^1(\mathbf{C})\times \mathbf{Z}/nk\mathbf{Z}
$$ 
where $k$ is a suitable integer $k$.