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?