There is an exercise in my course Elliptic Curves and I am not sure if I am doing it right. The question is as follows: Let $E$ be the elliptic curve over $\mathbb{F}_2$ given by $Y^2+Y=X^3$.
(a) Compute the dual of its Frobenius endomorphism
(b) Compute the degree of Frob + [1]
For (a) I know there is a theorem that says that the Frobenius endomorphism $\phi$ satisfies \begin{align} \phi^2-a\phi+q=0 \end{align} where $a=q+1-\#E(\mathbb{F}_q)$. So here we would have $a=0$ so $\phi^2+2=0$. Then we would have $\hat{\phi}=\phi$ since $\phi \phi=[-2]$. Is this the right approach?
For (b) I would say that we have $deg(\phi+1)=(\phi+1)(\hat{\phi+1})=(\phi+1)(\phi+1)$. But if I calculate this I don't get an integer, so I suppose I am doing something wrong here, but what?
You are almost right: since $\phi\widehat{\phi}=2$ by definition of dual and $\phi^2=-2$, you have $\widehat{\phi}=-\phi$. Now you should get the right result in (b), your argument is correct!