I'm trying to understand the idea of isomorphim classes of $\mathfrak{sl}_2(\mathbb{C})$-modules. for example, how can i find the solution for this question:
How many $\mathfrak{sl}_2(\mathbb{C})$ - modules $M$ exist such that $\operatorname{dim}(M) = 6$, up to isomorphism?
I’d tried to work with the fact that finite dimensional $\mathfrak{sl}_2(\mathbb{C})$-modules are completely reducible.