TITLE: Question on the Sato-Tate conjecture
QUESTION [6 upvotes]: Let $E$ be an elliptic curve over $\mathbb{Q}$ without CM. For a good prime $p$, define $\theta_{E}(p)$ by
$$\cos\theta_{E}(p)=\frac{p+1-N_{p}(E)}{2\sqrt{p}}\quad (0\leq \theta_{E}(p)\leq \pi).$$
I wonder if the function $\theta_{E}$ is injective?

REPLY [12 votes]: No. If $E_p$ is a supersingular elliptic curve and $p>3$ then trace of Frobenius on $E_p$ is zero, so $\theta_E(p)=\pi/2$.
By a result of Elkies any elliptic curve over $\mathbb{Q}$ has supersingular reduction in infinitely many primes, so for infinitely many $p$ this function takes the same value.