Elliptic numerical ranges of 3 × 3 companion matrices

A companion matrix is determined by the zeros of it’s characteristic polynomial. We determine the location of the zeros which yields an elliptic numerical range. In particular we show that given any two complex numbers z1 and z2 there exists a third complex number z3 such that the companion matrix of the polynomial (z-z1)(z-z2)(z-z3) will have elliptic numerical range with foci at z1 and z2.