Finding torsion subgroups of elliptic curves over finite fields.
Given $y^2=x^3+x+1$ over $F_3$ I need torsion subgroup of $E$
$E$ is either trivial or isomorphic to $\mathbb Z_3$
The points $(1,0),(-1,0),(0,0)$ are each of order $2$, so useless, but the point $(3,1)$ has order $4$ i.e. $4(3,1)=(\infty,\infty)$, Is there a possibility to combine this point with the others (or with itself, I cannot see it now) to get an element of order $3$ ?