A modular equation of 19th degree of Dedekind’s $\eta$ function.

Question

Regarding the Post Additional values of Dedekind's $\eta$ function in radical form I wrote the equation that has as root the value $\frac{\eta(19i)}{\eta(i)}$ that is missing.

Can someone help me solve (in radical form) the following equation, whose solution is the value of Dedekind’s modular $\frac{\eta(19i)}{\eta(i)}$ function?

$x^{40}-\frac{4}{19}x^{36}+\frac{138}{ 19^{3}}x^{32}-\frac{154}{19^{4}}x^{28}+\frac{41035}{19^{7}}x^{24}-\frac{359820}{19^{9}}x^{20}+\frac{1743935}{19^{11}}x^{16}-\frac{4798430}{19^{13}}x^{12}+\frac{16921266}{19^{15}}x^{8}+\frac{422140}{19^{17}}x^{4}-\frac{1}{19^{19}}=0$

where

$$x=\frac{\eta(19i)}{\eta(i)}.$$

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.

Answer 1

After substitution $x^4\to x,$ we get a 10-degree polynomial that factors over the rationals extended with $\sqrt{19}$ and a root of a solvable quintic $$\!\!\!\!\!\!\!\!\!\!\small 322687697779 \,z^5-67934252164 \,z^4+5974826887 \,z^3-229104318 \,z^2+3132036 \,z+216=0,\tag{$\small\spadesuit$}$$ so your 40-degree root can be represented in radicals. Let $$\small\begin{align} \alpha &=30 \sqrt{198616747730+65513019062 \sqrt{5}},\\ \beta &=30 \sqrt{198616747730-65513019062 \sqrt{5}},\\ \eta &=\sqrt[5]{11410567+2790935 \sqrt{5}+\alpha\;}+\sqrt[5]{11410567+2790935 \sqrt{5}-\alpha\;}\\ &+\sqrt[5]{11410567-2790935 \sqrt{5}+\beta\;}-\sqrt[5]{2790935 \sqrt{5}-11410567+\beta\;},\,\text{and}\\ \gamma& =8-\left(\frac{2}{19}\right)^{4/5} \eta, \end{align}$$ then $$\small\begin{align}&\!\!\!\frac{\eta(19i)}{\eta(i)}\\ &=\frac{\sqrt[4]{100680000 +7361892000{\tiny\text{ }}\gamma+76992000 \sqrt{19}\,\gamma -1888138300{\tiny\text{ }}\gamma ^2+145028140{\tiny\text{ }}\gamma ^3-4533799{\tiny\text{ }}\gamma ^4}}{20\cdot 2^{3/4} \cdot19^{3/8}\cdot \sqrt[4]{1203\,}}. \end{align}$$ I found the quintic $\small(\spadesuit)$ using my computer program that uses a combination of brute force and heuristics to find an extension of rationals in which a given polynomial can be factored.

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Mathematica expression for the root:

With[{α = 30 Sqrt[198616747730 + 65513019062 √5], 
      β = 30 Sqrt[198616747730 - 65513019062 √5]},
  With[{η = (11410567 + 2790935 √5 + α)^(1/5) 
          + (11410567 + 2790935 √5 - α)^(1/5) 
          + (11410567 - 2790935 √5 + β)^(1/5) 
          - (2790935 √5 - 11410567 + β)^(1/5)}, 
    With[{γ = 8 - (2/19)^(4/5) η}, 
       (100680000 + 7361892000 γ + 76992000 √19 γ 
         - 1888138300 γ^2 + 145028140 γ^3 - 4533799 γ^4)^(1/4)
            /(20 2^(3/4) 19^(3/8) 1203^(1/4))]]]