Upper Bounds for the Essential Dimension of $E_7$ This paper gives a new upper bound for the essential dimension and the essential 2-dimension of the split simply connected group of type $E_7$ over a field of characteristic not 2 or 3. In particular, $\operatorname{ed}(E_7) \leq 29$, and $\operatorname{ed}(E_7;2) \leq 27$. Keywords:$E_7$, essential dimension, stabilizer in general positionCategories:20G15, 20G41