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Should it not the resulted Elliptic Curve be defined over $F_{p^{2}}$ instead of $F_{p^{8}}$

Hello, I am trying to compute the quadratic twist for an example of an Elliptic curve defined over a GF(p^8) Field: With

p=3351951982486667453837338848452726606028033606935065896469552348842908133596080151967071453287452469772970937967942438575522391344438242727636910570385409

and an Elliptic curve defined as:

E = ElllipticCurve(GF(p),[1,0])

given the extensions:

F2.<i> = GF(p^2, modulus=x^2 + 11)
F4.<j> = GF(p^4, modulus=x^4 + 11)
F8.<k> = GF(p^8, modulus=x^8 + 11)

I am trying to compute a twist of the elliptic curve defined over F8, of the twist equation form: y^2=x^3+a w^4 x, while a=1, and w satisfies the following: $w\in F_{p^8}$ and $w^4 \in F_{p^2}$, $w^2 \in F_{p^4}$ and $w^3 \in F_{p^{8}} \setminus F_{p^{4}}$

Is there any sage command that would help with this problem ?

Another way but I didn't know how to write it in sage: {1,w,w^2,w^3} are the basis of $F_{p^8}$ as a vector space over $F_{p^{2}}$.

Thanks in advance.