# Compute elements of n-torsion group of elliptic curve over finite field

Suppose $F_q$ is a prime field and $E/F_q$ is an elliptic curve over that field with order $k= n\cdot ...$ and assume that $E$ has embedding degree $l$. Then the $n$-torsion group of $E$ is in $F_{q^l}$. Now assume that $n$ and $l$ are reasonably small, such that the $n$-torsion group contains only a few elements and can be listed.

How can I compute that group and list the elements?

edit retag close merge delete

Ok, then take say $E$ to be defined over $F_{43}$ by $y^2=x^3+6$. This is a very simple BLS6 curve (hence has embedding degree 6). It has a prime oder subgroup of order 13 and I want to list the entire 13-torsion.