I am interested in the zeros of $j''(z)$, where $j:\mathbb{H}\rightarrow\mathbb{C}$ is the classic modular function. Specifically I am interested in knowing if the zeros of $j''$ are algebraic over $\mathbb{Q}$, or, even better, quadratic over $\mathbb{Q}$.
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I doubt they are. Taking the derivative with respect to $\tau$ does not preserve the algebraic nature of modular forms. Why are you interested in this question? – Bruno Joyal Apr 01 '16 at 17:29
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Zeros of $E_4$ are also zeros of $j''$ because $j$ has the zeros of $E_4$ tripled. That part is easy. The remaining zeros are those of $-E_2E_4E_6+3E_4^3+4E_6^2$. Hmmm... – ccorn Apr 02 '16 at 18:54