Torsion point of an elliptic curve ($E[n]$)

The $n$-torsion subgroup $E[n]$ is the kernel of multiplication by $n$. When the characteristic does not divide $n$, it is isomorphic over an algebraic closure to $(\mathbb{Z}/n\mathbb{Z}{)}^2$.

The $n$-division field is obtained by adjoining to $K$ the coordinates of every point of $E[n]$. Over a finite field its degree is the order of Frobenius acting on $E[n]$.

The Weil pairing is a bilinear, alternating, nondegenerate pairingFor an isogeny $\varphi$ and its dual, it satisfies $e_n(\varphi P,Q)=e_n(P,\hat{\varphi }Q)$ whenever both sides are defined.