I have a question regarding dual isogenies. I read an example in Silverman's book about elliptic curves and am wondering something about this example. We have $\zeta$ as a primitive cube root of unity. Then the elliptic curve $C: y^2=x^3+1$ has complex multiplication: \begin{align*} \phi(x,y)=(\zeta x, -y) \end{align*} Now it is clear to me that we have $\phi^3(P)=-P$ and $\phi^6(P)=P$. But what is now the dual isogeny of $\phi$? Since we could take $\hat{\phi}=\phi^2$ and have $\phi \phi^2= [-1]$ or we could take $\hat{\phi}=\phi^5$ and get $\phi \phi^5 = [1]$. How do I know which is the right one?
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By the definition of the dual isogeny, you have that $\hat{\phi}\phi = [\deg\phi]$.
In this case, $\deg\phi = 1$ (and besides, the degree of an isogeny cannot be negative). Thus $\hat{\phi} = \phi^5$.
Andrea
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