Why does the derivative of the j-Invariant have only two zeros?

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In Tom Apostols Proof of Picards Theorem [Modular Functions and Dirichlet Series in Number Theory, Thm. 2.10] he uses the Propertie of J' [Derivative of the J-Invariant] being only zero at $\rho$ and i and I get that those are in fact Zeros i just havent found an Argument that would suggest that those are the only ones.