Solution
= Solution
Direct enumeration of the solutions of $y^2+y=x^3-4x+2$ gives
$$
\begin{array}{c|c|c}
p&\#\{(x,y)\in\mathbb F_p^2:y^2+y=x^3-4x+2\}&\#\widetilde E(\mathbb F_p)\\ \hline
3&3&4\\
5&8&9\\
7&9&10
\end{array}
$$
after adjoining the point at infinity. Therefore
$$
\boxed{\#\widetilde E(\mathbb F_3)=4,
\quad\#\widetilde E(\mathbb F_5)=9,
\quad\#\widetilde E(\mathbb F_7)=10.}
$$