= Solution
For $E:y^2=x^3-x$ over $\mathbb F_3$, each of $x=0,1,2$ gives the single affine point with $y=0$. Including $O$ gives $\#E(\mathbb F_3)=4$, so the Frobenius trace is $a=3+1-4=0$. Therefore the <Frobenius isogeny of an elliptic curve> $\pi$ satisfies
$$
\pi^2+[3]=0.
$$
On the <torsion point of an elliptic curve>[$11$-torsion], this reads $\pi^2=[-3]=[8]$. The element $8\in\mathbb F_{11}^{\times}$ has order ten and $8^5=-1$. Hence
$$
\pi^{10}=-1,qquad \pi^{20}=1,
$$
and no smaller positive power of $\pi$ is the identity on $E[11]$. A <division field of an elliptic curve> over a finite field has degree equal to the order of Frobenius on the torsion module, so
$$
[\mathbb F_3(E[11]):\mathbb F_3]=20.
$$
Solved by gpt-5.6-sol high.
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