How to solve a quintic polynomial using elliptic functions with Mathematica?

Question

I followed the exact steps from this forum post to solve quintic polynomials of the form:

$x^5 - x + d$

But I got a different answer in number form from Mathematica than the original quintic polynomial root.

Am I doing something wrong?

Also, where it says, $(k^2)^{1/8}$, isn't that supposed to be $k^{1/4}$?

Answer 1

d = RandomInteger[{-100, 100}];
Print["\nEquation: x^5 - x ", If[d < 0, " - ", " + "], Abs[d], " = 0\n"];
Print["Ordinary solution:"];
Print[x /. (x^5 - x + d // NSolve) // Sort, "\n"];
Print["Solution with Dedekind Eta:"];
k = Tan[ArcSin[16/(25 Sqrt[5] d^2)]/4];
p = I EllipticK[1 - k^2]/EllipticK[k^2];
S[J_] = Exp[2 Pi I/8]^J Sqrt[2] DedekindEta[(p + 2 J)/10] DedekindEta[4 (p + 2 J)/10]^2/DedekindEta[2 (p + 2 J)/10]^3 ;
S5 = Sqrt[2] DedekindEta[5 p/2] DedekindEta[10 p]^2/DedekindEta[5 p]^3 ;
Sign[d] (1/(2 5^(3/4))) ((k^2)^(1/8)/Sqrt[k (1 - k^2)]) (S[0] + S5) (S[1] + I S[4]) (I S[2] + S[3]) // N