Sylvester versus Gundelfinger

Let $V_n$ be the ${ m SL}_2$-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 $R = mathbb{C}[V]^{{ m SL}_2}$ of polynomial functions on $V$ invariant under the action of ${ m SL}_2$ equals 63. This settles a 143-year old question.