= 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 $y^2+y=0$. Thus
$$
\boxed{\#\widetilde E(\mathbb F_2)=5,\qquad\#\widetilde E(\mathbb F_3)=7.}
$$
Both reductions are nonsingular by the discriminant calculation, so these are the point counts of the reduced <elliptic curves>, with their <group> laws.
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