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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?

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