Does roots of L-polynomial of curve being roots of unity imply the curve is supersingular?

Question

If $C$ is a supersingular curve over $\mathbb F_q$, then $\frac1{\sqrt q}$ of roots of $L$ function are roots of unity.

What about converse? If $\frac{1}{\sqrt q}$ of roots of $L$ polynomial are roots of unity, then is $C$ supersingular?

---

Or,

Is there a curve $C$ which attains maximum or minimum Weil bound over some extension $\mathbb F_{q^n}$ but $C$ is not supersingular?