I get multiple sets of elliptic curves (EC) and need to prove that the set either forms or doesn't form a group with the point at infinity. How do I do that?
An example
Does the set $E\left(\mathbb{F}_{23}\right)=\left\{(x, y) \in \mathbb{F}_{23} \times \mathbb{F}_{23}: y^{2}=x^{3}+2 x-25\right\}$ form a group with the point at infinity?
I think all you have to do is the following:
$i)$ check that the field has neither characteristic $2$ nor $3$ ($\mathrm{char}(\mathbb{F}_{23})=23$);
$ii)$ check that the curve is non-singular ($4\cdot \color{blue}{2}^3 + 27\cdot (\color{blue}{-25})^2 \not\equiv 0\ \mathrm{ mod}\ 23$) .