It is continuation of this question: explict form of the equation of elliptic curve

Let $p$ is prime and $p = 3 ($mod $4)$. $q = p^n$. It is easy to see that $E: y^2 = x^3 + x$ has $1 \pm 2q + q^2$ rational points over $\mathbb{F}_{q^2}$.

It is excerpt from "Constructing supersingular elliptic curves" of Reinier Broker(http://www.math.brown.edu/~reinier/supersingular.pdf) for $p \not= 1$ mod $3$, we can twist by a primitive sixth root of unity $\zeta_6 ∈ \mathbb{F}_{q^2}$ to obtain curves with trace of Frobenius $\pm 2q$ and $\pm q$ (I changed some for saving my designation).

So how it can be done? Or can you give a reference on material about twist of elliptic curves over finite field?


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