Minimal height companion matrices for Euclid polynomials

We define Euclid polynomials $E_{k+1}(\lambda )=E_k(\lambda )(E_k(\lambda )-1)+1$ 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 $E_k(\lambda )=\det (\lambda \mathbf{I}-{\mathbb{E}}_k)$, 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.