Elliptic Curves over extension fields

Suppose that $p$ is a prime number, and $F_p$ its associated prime field. We know that the fields $F_{p^m}$ are extensions of $F_p$ in the sense that $F_p$ is a subfield of $F_{p^m}$ . This implies that we can extend the affine plane that an elliptic curve is defined on by changing the base field to any extension field. To be more precise, let $E(F) = \{(x, y) ∈ F × F | y^2 = x^3 + a · x + b\}$ be an affine Short Weierstrass curve, with parameters $a$ and $b$ taken from $F$. If $F'$ is an extension field of $F$, then we extend the domain of the curve by defining $E(F')$ as follows:

$$E(F') = \{(x, y) ∈ F' × F' | y^2 = x^3 + a · x + b\}.$$

While we did not change the defining parameters, we consider curve points from the affine plane over the extension field now. Since $F ⊂ F'$, it can be shown that the original elliptic curve $E(F)$ is a sub-curve of the extension curve $E(F')$.

Sage example:

# Bn254
p = 21888242871839275222246405745257275088696311157297823662689037894645226208583
F_p = GF(p)
(a, b) = (0, 3)
E = EllipticCurve(F_p,[a, b])

m = 12
F_pt.<t> = F_p[]
P_MOD = F_pt.irreducible_element(m)
F_pm.<t> = GF(p^m, name='t', modulus=P_MOD)
E_ext = EllipticCurve(F_pm, [a, b])
E_ext