Minimal height companion matrices for Euclid polynomials

We define Euclid polynomials $E_{k+1}(\lambda) = E_{k}(\lambda)\left(E_{k}(\lambda) - 1\right) + 1$ and $E_{1}(\lambda) = \lambda + 1$ in analogy to Euclid numbers $e_k = E_{k}(1)$. We show how to construct companion matrices $\mathbb{E}_k$, so $E_k(\lambda) = \operatorname{det}\left(\lambda\mathbf{I} - \mathbb{E}_{k}\right)$, of height 1 (and thus of minimal height over all integer companion matrices for $E_{k}(\lambda)$). We prove various properties of these objects, and give experimental confirmation of some unproved properties.