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 $$A+B+C$$.

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