I have several questions on Hilbert scheme of Gushel-Mukai varieties. Let $X$ be a Gushel-Mukai fourfold and let $\mathcal{H}_3$ be Hilbert scheme of twisted cubics. I was wondering what is the dimension of $\mathcal{H}_3$? If I consider the locus of reducible twisted cubic, say a line and a conic. Since Hilbert scheme of lines is of dimension $3$ and Hilbert scheme of conics is of dimension $5$, then dimension of such locus is $3+5-1=7$. But I am not sure the dimension of locus of smooth twisted cubics?

The second question, if $Y$ is a Gushel-Mukai threefold, is there any space of rational curves or space of elliptic curves on $Y$ such that its dimension is $6,7,8$?