432 Is Special

Let S S be the set of rational numbers x x such that x3432 x^3-432 is the square of a rational number.

If S S is infinite, enter 1 -1 . If S S is finite, enter the sum of the elements of S. S.

Here is a very helpful hint:
If y2=x3432, y^2=x^3-432, let u=36y6x,v=36+y6x u = \dfrac{36-y}{6x}, \, v = \dfrac{36+y}{6x} . What is u3+v3 u^3+v^3?

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