Division polynomials (${\psi }_n$)

For an elliptic curve $E:y^2=x^3+Ax+B$ in characteristic different from two and three, division polynomials are elements of its coordinate ring encoding multiplication and torsion points of an elliptic curve. The initial terms includeThe recurrences for higher $n$ yieldFor odd $n$, ${\psi }_n$ is a polynomial in $x$; for even $n$, it has a factor $2y$, and ${\psi }_n/(2y)$ is a polynomial in $x$. Thus the phrase “roots of the division polynomial” must specify whether it means points on the curve or roots in the $x$-line. When $n$ is prime to the characteristic, its zero points are exactly the nonidentity $n$-torsion points of an elliptic curve; for odd $n$, the two opposite points share an $x$-root.^[ref]