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Regarding the Post Additional values of Dedekind's $\eta$ function in radical form I wrote the equation that has as root the value $\frac{\eta(23i)}{\eta(i)}$ that is missing.

Can someone help me solve (in radical form) the following equation, whose solution is the value of Dedekind’s modular $\frac{\eta(23i)}{\eta(i)}$ function?

$$x^{48}+\frac{684}{23^{5}}x^{36}-\frac{2496}{23^{7}}x^{32}+\frac{10944}{23^{9}}x^{28}+\frac{3826738}{23^{11}}x^{24}-\frac{31577472}{23^{13}}x^{20}+$$ $$\frac{785460096}{23^{15}}x^{16}-\frac{2112004548}{23^{17}}x^{12}+\frac{4240221504}{23^{19}}x^{8}+\frac{18998208}{23^{21}}x^{4}-\frac{1}{23^{23}}=0$$

where

$$x=\frac{\eta(23i)}{\eta(i)}.$$

This equation comes from the work of L. Kiepert and specializes for the value reported in the title of the application. My intent is to find the solution in closed form.

giuseppe mancò
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1 Answers1

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After substitution $x^4\to x,$ we get a 12-degree equation that factors into cubics over the rationals extended with $\sqrt{23}$ and $\sqrt2\cdot\sqrt[4]{23\,}$. Solving the cubic equation, we get

$$\small\begin{align} \!\!\frac{\eta(23i)}{\eta(i)}&=\\ &\!\!\!\!\!\!\sqrt[4]{\frac{2\cdot 2^{2/3} \sqrt[3]{\alpha }+\sqrt[3]{\frac{2}{\alpha }} \left(360 \sqrt{23}+9 \sqrt{2} \cdot23^{3/4}-1426-525 \sqrt{2}\, \sqrt[4]{23}\right)-8 \sqrt{23}+6 \sqrt{2}\,\sqrt[4]{23} \left(3+\sqrt{23}\right)}{6348}}, \end{align}$$ where $$\small\alpha =32 \sqrt{23} \left(38-9 \sqrt{3}\right)+2484 \left(9 \sqrt{3}-4\right)-9 \sqrt{2}\, \sqrt[4]{23} \,\left(717-454 \sqrt{3}+139 \sqrt{23}+62 \sqrt{69}\right).$$

Vladimir Reshetnikov
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