Let E be an elliptic curve over $F_p$. Suppose that its j invarient is not supersingular and that $j\neq 0 $ or 1728.
Then the modular polynomial $\Phi_l(j,T)$ has a zero $\tilde{\jmath} \in \mathbb{F}_{p^r}$ if and only if the kernel $C$ of the corresponding isogeny $E \mapsto E/C$ is a one-dimensional eigenspace of $\phi^r_p$ in $E[l]$, with $\phi_p$ the Frobenius endomorphism of $E$.
In the proof: If $C$ is an eigenspace of $\phi_p^r$, it is stable under the action of the Galois group generated by $\phi_p^r$. Therefore the isogeny $E \mapsto E/C$ is defined over $\mathbb{F}_{p^r}$ and the $j$ invariant $\tilde{\jmath}$ of $E/C$ is contained in $\mathbb{F}_{p^r}$ . Counting points on eliptic curve over finite field page no:236.
Can someone help me to understand the prrof, I dont know the relation between eigenspace of $\phi_p^r$ and Galois group generated by $\phi_p^r$. Please help me with a small example or, theorem providing this.