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

asked 2021-06-19 14:49:48 +0200

Planet Macro gravatar image

updated 2021-08-11 14:09:05 +0200

FrédéricC gravatar image

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?

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Welcome to Ask Sage! Thank you for your question!

slelievre gravatar imageslelievre ( 2021-06-19 16:04:50 +0200 )edit

Ideally, provide an example of defining all the pieces to get to the actual question.

slelievre gravatar imageslelievre ( 2021-06-19 16:05:21 +0200 )edit

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.

Planet Macro gravatar imagePlanet Macro ( 2021-06-19 16:43:21 +0200 )edit