### The CM Exception

Let us explore the Sato-Tate conjecture for this elliptic curve:
Obviously, those numbers do not satisfy the sin

What is going on? The explanation is that the above elliptic curve belongs to a small exceptional class called CM elliptic curves, where CM stands for complex multiplication. For this class, the Sato-Tate conjecture is not supposed to hold.

^{2}law. Indeed, some elementary number theory shows that if the prime number p is of the form 4n+3, then the equation y^{2}= x^{3}- x has exactly p solutions mod p, so that a_{p}=0. This means φ_{p}=π/2 and explains the prominent spike in the middle.What is going on? The explanation is that the above elliptic curve belongs to a small exceptional class called CM elliptic curves, where CM stands for complex multiplication. For this class, the Sato-Tate conjecture is not supposed to hold.

Finally, check out the notes and references on page 5.