If I have the following Elliptic Curve:
$$E: y^2 \equiv x^3 + 2x + 2 \pmod{17}$$
How can I calculate the number of points on this elliptic curve $E$ ?
and how can I invest the following law which calculates the Number of Points on an Elliptic curve NPE:
$$NPE=1+p+\sum_{x=1}^{p-1} \frac{f(x)}{p}$$
Let $\;f(x) := x^3+2x+2\;$ be a cubic. An Elliptic curve $\;E: y^2 \equiv f(x) \pmod{p}\;$ is defined as the set of points $\;(x,y)\;$ which satisfy the congruence and the point at infinity $O$, on which an addition operation is defined which makes it a group with the additive identity being the point at infinity $O$.
Thus the number of points $\;NPE=1+|\{(x,y):y^2=f(x)\}|\;$ where we work modulo $p$. The number of solutions of the quadratic $\;y^2=f(x)\;$ is given by the Legendre symbol as $\;1\!+\!\left({f(x) \over p}\right)\!.$ Thus
$$\;NPE=1+\sum_{x=0}^{p-1}1+\left({f(x) \over p}\right) = 1 + p +\sum_{x=0}^{p-1}\left({f(x) \over p}\right).\;$$