Let $E/K$ be an elliptic curve and $m$ be a positive integer such that char$(K)\nmid m$. I know that the Weil pairing defined on elliptic curves is a bilinear & non-degenerate pairing. However is the pairing perfect? Do we have,
$$E[m]\cong \text{Hom}(E[m],\mu_m)$$ If so, how?