Sylvester versus Gundelfinger

Let $V_n$ be the $\mathrm{SL}_2$-module of binary forms of degree $n$ and let $V=V_1\oplus V_3\oplus V_4$. We show that the minimum number of generators of the algebra $R = \mathbb C[V]^{\mathrm{SL}_2}$ of polynomial functions on $V$ invariant under the action of $\mathrm{SL}_2$ equals 63. This settles a 143-year old question.