The differential invariants of $S{L}_2({\mathbb{F}}_3)$ acting on trace-free matrices over ${\mathbb{F}}_3$

Let $M$ denote the vector space of $2\times 2$ matrices with coefficients in ${\mathbb{F}}_3$ and trace zero. Let $G=S{L}_2({\mathbb{F}}_3)$. Then $G$ acts on $M$ via conjugation. Let $R=(S(M^{\ast })\otimes \mathrm{\Lambda }(M^{\ast }))$ be the algebra of differential forms on $M$. We compute a minimal generating set for $R^G$ as a commutative-graded algebra. In doing so we utilise the theory of Cohen-Macaulay modules and results in the theory of covariants.