Verifying $\ker( \phi_q -1) = E(F_q) $ by an example

Question

I want to verify $\ker(1- \phi_q ) = E(F_q) $ by an example where $\phi_q$ is the Frobenius map $(x^q,y^q)$. If we take $E: y^2= x^3 + 3x +2$ over $F_5$, then the points on it are ${(1,1),(1,4),(2,1),(2,3),(4,0),\infty}$. But I do not see how these points are in $ker(1- \phi_q )$? If $\alpha = 1- \phi_5$, can we say $\alpha = (x-x^5,y-y^5)$? I appreciate any help!