A normal with slope $\frac { 1 } { \sqrt { 6 } }$ is drawn from the point $( 0 , - \alpha )$ to the parabola $x ^ { 2 } = - 4 a y$, where $a > 0$. Let $L$ be the line passing through $( 0 , - \alpha )$ and parallel to the directrix of the parabola. Suppose that $L$ intersects the parabola at two points $A$ and $B$. Let $r$ denote the length of the latus rectum and $s$ denote the square of the length of the line segment $AB$. If $r : s = 1 : 16$, then the value of $24 a$ is $\_\_\_\_$ .