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I'm looking at the elliptic curve $C:={\cal Z}(XY^2+ZX^2+YZ^2)$ in the field $k:=\overline{\mathbb{F}_2}$. I want to prove that this curve has 9 inflection points. Since the characteristic of $k$ is 2, I cannot use the Hessian determinant, which is always zero in this case. I have also shown that ${\cal Z}(X)\cap C$, ${\cal Z}(Y)\cap C$ and ${\cal Z}(Z)\cap C$ aren't inflection points. Can anybody help me?

Thanks in advance

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