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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 $\frac{\eta(11i)} {\eta(i)}$ that is missing.

Can someone help me solve (in radical form) the following equation, whose solution is the value of Dedekind's modular $\frac{\eta(11i)} {\eta(i)}$ function?

$$x^{24}-\frac{90}{11^{5}} x^{12}+\frac{480}{11^{7}}x^{8}+\frac{288}{11^{9}} x^{4}-\frac{1}{11^{11}}=0$$

where

$$x=\frac{\eta(11\ i)}{\eta(i)}.$$

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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Using Mathematica I found a solution. Let $\ p := 11,\ $ $s := \sqrt{p},\ $ $x := \eta(p\ i)/\eta(i),\ $ $x_0 := (p\,x^2)^2,\ $ $f(x) := x^3 -4sx^2 + (88-12s)x + (33-10s).\ $ Then $\ f(x_0) = 0.\ $ Solve for $\ x_0\ $ using $\ c_1 := 22-9s,\ $ $c_2 := 21365-6324s,\ $ $c_3 := -3861+1490s,\ $ $r_2 := 9s\sqrt{3c_2},\ $ $r_3 := \sqrt[3]{(c_3-r_2)/2}.\ $ Then $\ x_0 = (4s+4c_1/r_3-r_3)/3 \approx 0.003452370313107857. $

Somos
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    @OscarLanzi They are two different numbers. One is $x$ root of degree 24 polynomial. The other is $x_0 = (p,x^2)^2$ root of a cubic. I voted to close by mistake. – Somos Apr 29 '19 at 02:01
  • So I'm looking at $121x^4$. Seems to work. – Oscar Lanzi Apr 29 '19 at 02:06
  • Thank you! I posted $\eta(11i)$ https://math.stackexchange.com/q/3101157/additional-values-of-dedekinds-eta-function-in-radical-form. – giuseppe mancò May 02 '19 at 11:32