I am trying to understand the CM method for elliptic curves. Suppose we fix a discriminant $D<0$ and a prime $p$. In the CM method, we look for integer solutions $(t,y)$ to the norm equation $4p = t^2 -Dy^2$. If these solutions exist, then we can construct an elliptic curve over $\mathbb{F}_p$ with $p+1 \pm t$ rational points. What I find it hard to understand is that this step only involves studying the Hilbert polynomial $H_D$ modulo $p$, and we recover a curve with trace $\pm t$ (and not some other $t'$ that might satisfy the norm equation).

My question is, is the solution $(t,y)$ to the norm equation $4p=t^2-Dy^2$ unique if we fix $D$ and $p$ (and assume $t,y>0$)? If so, why?