On the degree of a modular map

Let $X$ be a general cubic hypersurface in ${\mathbb{P}}^4$. If $x\in X$ is a general point there are exactly six distinct lines in $X$ passing through $x$, that lie on the rank 3 quadric cone with vertex $x$ of lines that have intersection multiplicity at least 3 with $X$ in $x$. So there is a natural rational map $X\text{dasharrow}{\mathcal{M}}_2$. In this paper we compute its degree to be 2074320.