Understanding quotient classes in finite field - explicit example

Question

I have a little doubt.
How do I find representants for classes of the quotient $$\big( \mathbb{F}_{11}[X]/(X^3-4x^2+23) \big)^{\ast} /\Big( \big( \mathbb{F}_{11}[X]/(X^3-4x^2+23) \big)^{\ast}\Big)^7 $$ I think that such quotient should be isomorphic to $\mathbb{Z}/7\mathbb{Z}$ using the fact that $\mathbb{F^\ast_{11^3}} \cong \mathbb{Z}/1330\mathbb{Z}$ and raising to the 7-th power in the LHS is equivalent to multiplication by 7 in the RHS. But this is not particularly helpful in order to find explicitly representative in the 1st quotient