I want to find the order of elliptic curve over the finite field extension $\mathbb{F}_{p^2}$, where $E(\mathbb{F}_{p^2}):y^2=x^3+ax+b $
I am using the method illustrated by John J. McGee in his thesis 2006. Where $\#E(\mathbb{F}_{p^n})=p^n+1-(s_{n})$, with, $s_0=2$, $s_1=t$ and $s_{n+1}=t s_n - ps_{n-1}$.
Finding $t$ is easy by using Weil's theorem, where $\#E(\mathbb{F}_p)=p+1-t.$
McGee had put $s_0=2$, but he did not say why, nor he gave a reference. Therefore my question is: What is the condition to determine $s_0$? Is it supposed to be $2$ at all cases? And Why? I am asking this, because I worked on few examples where I found the number of points is not the same order when $s_0=2$.
Worth to say, the method that I am using to find the points of $E(\mathbb{F}_{p^2})$ is the same method to find the point of elliptic curve over $\mathbb{F}_p$.
Thank you in advance.