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I already asked a closely related question on MSE.

Considering the modular lambda function, the values of $ \lambda^{*}(n) $ for some integers are given on here.

Is there a way to calculate the coefficients of the minimal polynomials of each $ \lambda^{*}(n) $ ?

Here you have the minimal polynomials for the first $10$ positive integers, found with Mathematica

\begin{array}{|c|c|} \hline n & P_n(x) \\ \hline 1 & 2x^2-1 \\ \hline 2 & x^2+2x-1 \\ \hline 3 & 16x^4-16x^2+1 \\ \hline 4 & x^2-6x+1 \\ \hline 5 & 16x^{8} - 32x^{6} + 88x^4 - 72 x^2 + 1 \\ \hline 6 & x^{4} + 12x^{3} +2x^2 - 12x +1 \\ \hline 7 & 256x^4 - 256x^2 + 1 \\ \hline 8 & x^{4} - 20x^{3} - 26x^2 - 20x +1 \\ \hline 9 & 16x^{8} - 32x^{6} + 792x^4 - 776x^2 + 1 \\ \hline 10 & x^{4} + 36x^{3} + 2x^2 - 36 x + 1 \\ \hline \end{array}

Ulissex
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user967210
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    There is way to compute the modular equation of degree $p$ for the modular function $\lambda$ but its mostly theoretical. Calculating actual coefficients is a computational problem and further if $P(x, y) =0$ is the modular equation we need to substitute $x=\lambda, y=1-\lambda$ to get the desired polynomial equation for $\lambda$ as $P(\lambda, 1-\lambda)=0$. – Paramanand Singh Apr 30 '24 at 18:19
  • @ParamanandSingh thank you for the answer and sorry for the late reply. Could you please give me any reference? – user967210 May 19 '24 at 06:46
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    You should check Borwein's Pi and the AGM from page 119 onwards. – Paramanand Singh May 19 '24 at 15:32

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