constant term of division polynomials of elliptic curves

Question

I am mainly looking for a reference request, if it exists, to save some time looking for it.

Let $E/\mathbb Q$ an elliptic curve in Short Weierstrass form and $\phi_n,\omega_n,\psi_n$ the usual division polynomials ( $nP=( \frac{\phi_n(P)}{\psi_n(P)^2} , \frac{\omega_n(P)}{\psi_n(P)^3})$ ). If $E: y^2=x^3+Ax+B$ and you consider $\omega_n, \phi_n \in \mathbb Z[A,B,x]$ ( consider $n$ even), is there a formula for their constant term? i.e a formula for $\phi_n(0), \omega_n(0)$ in terms on $n,A,B,x$ (when considered as polynomials in terms of $x$)?

Answer 1

Given the elliptic curve $$ E: y^2=x^3+Ax+B $$ when $x=0$ then $y=\sqrt{B}.$ From the Wikipedia article Division polynomials $$ \phi_n = x \psi_n^2 - \psi_{n+1}\psi_{n-1} $$ and when $x = 0$ it simplifies to $$ \phi_n = -\psi_{n+1}\psi_{n-1}. $$ Also, the following is true $$ \omega_n = \psi_{2n}/(2\psi_n). $$ It think this is about as simple as it gets.