# Sextic twist maps to q Eigenspace of Frobenius

Let $E(p)$ be a Barretto-Naehrig elliptic curve with r-torsion and embedding degree 12 and $E'$ a sextic twist with homomorphism $\psi$. How to show, that

• $E'$ has a unique r-torsion group
• $\psi$ maps the r-torsion of $E'$ to the Trace-0 subgroup of $E(p^{12})$
• @cryptostase Why should we delete the question? I'm no expert in that area, but it looks like a valid question to me. – CodesInChaos Sep 27 '15 at 10:03
• Since you found the answer yourself, you could post an answer referencing the paper and preferably giving a short summary of the proof idea. – CodesInChaos Sep 27 '15 at 10:05
• The proof can be found in the paper: The Eta Pairing Revisited, section 5 F. Hess, N. Smart, and Frederik Vercauteren – user28038 Sep 30 '15 at 16:36