Flat portions on the boundary of the numerical range of a 5 × 5 companion matrix

The number of flat portions on the boundary of the numerical range of $5 \times 5$ companion matrices, both unitarily reducible and unitarily irreducible cases, is examined. The complete characterization on the number of flat portions of a $5 \times 5$ unitarily reducible companion matrix is given. Also under some suitable conditions, it is shown that a unitarily irreducible $5 \times 5$ companion matrix cannot have four flat portions on the boundary of its numerical range. This gives a partial affirmative answer to the conjecture given in [3] for $n = 5$. Numerical examples are provided to illustrate the results.