Computing a sextic twist

Question

Let $(x,y) \in E'_{\mathbb{F}_{p^2}}$ be a point of the sextic twist.

I am currently trying to compute:

$\psi : (x, y) \leftarrow (\mu^2x,\mu^3y)$ with $\mu \in \mathbb{F}_{p^{12}}$ the root of $(Y^2 - \xi)$ with $\xi = 1+i \in \mathbb{F}_{p^{2}}$.

From the original curve over $\mathbb{F}_{p^{12}}$ (that is $y² = x³ + 2$) I found the twisted curve over $\mathbb{F}_{p^{2}}$ (that is $y² = x³ + (1-i)$) so I am currently able to easily find points in the twisted curve. However I don't really understand how I am supposed to compute $\mu$ in order to compute $\psi$ and get my point in $\mathbb{F}_{p^{12}}$.

Does anyone have any idea of how I can find $\mu$? I mean is there an algorithm that computes $\mu$? I know that $\mu^6=1+i$ but how do I compute $\mu$?

Thank you very much!

Answer 1

The correct answer depends on the way the towering is done (if any towering was done). I used the following towering:

$\mathbb{F}_{p^{2}}/(i^2+1)$

$\mathbb{F}_{p^{6}}/(y^3+\xi)$ with $\xi \in \mathbb{F}_{p^{2}}$

$\mathbb{F}_{p^{12}}/(x^2+\xi)$ with $\xi \in \mathbb{F}_{p^{2}}$

For this towering, $\mu^2=y$ and $\mu^3=x$.