We resolve the problem for the optimal hypercontractive constant $r_{p,q}(\mathbb{Z}_3)$ of the cyclic group $\mathbb{Z}_3$ for all $1< p<q< \infty$, by characterizing them via an explicit system of equations. As a consequence of our main result, for rational $p, q\in \mathbb{Q}$, the constants $r_{p,q}(\mathbb{Z}_3)$ are algebraic numbers which generally admit no radical expressions, since their often rather complicated minimal polynomials may have non‑solvable Galois groups. Our formalism relies on a key observation of the existence of nontrivial critical extremizers. This formalism can also be adapted to yield a resolution of the long-standing open problem of determining all optimal $(p,q)$-hypercontractive constants for biased Bernoulli random variables. The talk is based on a recent joint work with Jie Cao, Shilei Fan，Yong Han and Zipeng Wang.