Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 32 / 1 / ii / Solution

Over ${\mathbb{F}}_2$, both $x=0,1$ give $x^3-x=0$, and both $y=0,1$ give $y^2+y=0$. There are four affine rational points, together with the point $O$ at infinity. Over ${\mathbb{F}}_3$, each $x=0,1,2$ gives $x^3-x=0$, while precisely $y=0,2$ solve $y^2+y=0$. ThusBoth reductions are nonsingular by the discriminant calculation, so these are the point counts of the reduced elliptic curves, with their group laws.