Let $\phi: E \rightarrow E$ be the Frobenius morphism on an elliptic curve given by $(x,y) \mapsto (x^q, y^q)$ where $q$ is a prime power.
It induces a map on Tate module $T_l(E) = \varprojlim_{n} E[l^n]$, $E[l^n]$ is the set of all points of $E$ of order $l^n$ (here $n \geq 1$.) as below
$$ \phi_l : T_l(E) \rightarrow T_l(E) $$
It is given that $\phi_l$ is a $\mathbb{Z}_l$-linear map with characteristic polynomial say, $(T-\alpha)(T-\beta)$.
Now let $\phi^n : E(\mathbb{F}_{q^n}) \rightarrow E(\mathbb{F}_{q^n})$ be the map $(x,y) \mapsto (x^{q^n}, y^{q^n})$ and it similarly it induces a linear map on corresponding Tate module $\phi_l^n$.
My question: how do I find characteristic polynomial for $\phi_l^n$? The book says to triangularise/find Jordan normal form of it. The book says that it will be $(T- \alpha^n)(T-\beta^n)$.
But I'm not sure how to go on about finding this.
I'd be thankful if someone could help me out on this.