Let $V_n,V_m$ be the vector $\mathbb{C}$-spaces of the binary forms of degrees $n,m$ considered as usual $SL_2$-modules. Let $I_{n,m}=\mathbb{C}[V_n \oplus V_m]^{SL_2}$ and $C_{n,m}=\mathbb{C}[V_n \oplus V_m \oplus \mathbb{C}^2]^{SL_2}$ be the corresponding algebras of joint invariants and covariants. The transcendence degree of $I_{n,m}$ over $\mathbb{C}$ is $n+m-1.$

**Questions:**

What is the transcendence degree of $C_{n,m}$ over the subalgebra $I_{n,m}$?

What is the transcendence degree of $C_{n,m}$ over the quotient field of the subalgebra $I_{n,m}$?