The duplication formula for an elliptic curve over rationals:
$$y^2 = x^3 + ax^2 + bx + c$$
for the $x$-coordinate is given by:
$x$-coordinate of $2(x, y)=(x^4-2bx^2-8cx+b^2-4ac)/(4x^3+4ax^2+4bx+4c)$
Here $2(x, y)=(x, y)+(x,y)$ via the group law of addition of two points on the curve.
I am asking if there is a similar formula or a shape for the general case: $x$-coordinate of $n(x, y)$ where $n$ is a positive integer.
Can the duplicate formula be understood as an iterative sequence, i.e., replacing the $x$ coordinate of $n(x,y)$ by the $x$ coordinate of $(n+1) (x,y)$ ?
Best Answer
Yes, if $E$ is an elliptic curve, $P=(x,y)$ is a point on $E$, and $n\geq 1$, the coordinates of the point $Q=nP$ are given by the division polynomials $$nP = \left(\frac{\phi_n(x)}{\psi^2_n(x)},\frac{\omega_n(x,y)}{\psi^3_n(x,y)}\right).$$ The definition of the division polynomials can be found in this Wikipedia page, for example. These polynomials are defined recursively, so one can compute them easily (but they get very large as $n$ grows).