Newton's Sums with euler's number?

Algebra Level 3

$x+y+z=0\\ { x }^{ 2 }+{ y }^{ 2 }+{ z }^{ 2 }=e\\ { x }^{ 3 }+{ y }^{ 3 }+{ z }^{ 3 }={ e }^{ 2 }\\$

This set of equations is true for three complex numbers $x, y, z$ , where $e$ is Euler's Constant.

If $xyz=\frac { A }{ B } { e }^{ C }$ for positive integers $A,B,C$ with coprime $A,B$ , find $A+B+C$ .