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?



Question on the Sato-Tate conjecture
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.


