I am struggling to understand how to perform a sextic twist over a BN elliptic curve. This is what I understood so far:

Let's consider a BN elliptic curve: $$ E: y^2=x^3+b $$ And let's consider a point $Q \in E(F_{p^{12}}) $. What I would like to do is to "easily" go from/to $Q \in E(F_{p^{12}})$ to/from $Q' \in E'(F_{p^{2}})$, where $Q'$ is the sextic twist defined as $$ E': y^2=x^2+b/\xi $$ In fact I can use this isomorphism (I guess) to move from/towards $E$ and $E'$: $$ \psi: E'(F_{p^2}) \rightarrow E(F_{p^{12}}) $$ $$ \psi((x',y')) = (\xi^{1/3}x',\xi^{1/2}y') $$ $$ \psi^{-1}(x,y) = \left(\frac{x}{\xi^{1/3}},\frac{y}{\xi^{1/2}}\right) $$ My questions are:

  1. Is the isomorphism I wrote correct?
  2. Is $\xi$ an integer number in $F_p$?
  3. Can I choose any value I want for $\xi$? I guess that I should choose it so that $\xi^{1/3}$ and $\xi^{1/2}$ are integers, but I'm not sure...

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