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
$$
\boxed{\#\widetilde E(\mathbb F_3)=4,
\quad\#\widetilde E(\mathbb F_5)=9,
\quad\#\widetilde E(\mathbb F_7)=10.}
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 125 / 1 / b / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-125/1/b/ii/solution