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As yyyyyyy mentioned for counting number of points on elliptic curve over $\mathbb F_p$ we can use Elkies method. But for extension of fields use of this theorem make it so easy: Theorem : Let $E$ be an elliptic curve defined over $F_q$, and let $\#E(F_q ) = q +1−t$. Then $\#E(F_{q^n} ) = q^n + 1 − V_n$ for all $n ≥ 2$, where $\{V_n\}$ is the sequence ...

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