Appendix A: Reduction of the quintic to the Bring–Jerrard form
Let us first recast the quintic (43) as
$$\begin{aligned} x^{5}+a_{1} x^{4} + a_{2} x^{3}+ a_{3} x^{2}+a_{4} x +a_{5} = 0, \end{aligned}$$
(A1)
by defining \(a_{j}=\bar{m}_{5}^{-1}{\bar{m}_{5-j}}\). We now proceed with transforming Eq. (A1) to the principal quintic form that is missing the \(x^{4}\) and \(x^{3}\) terms, by means of the quadratic Tschirnhausen transformation
$$\begin{aligned} y = x^{2}+b_{1} x + b_{2}. \end{aligned}$$
(A2)
Applying a simple code in the software Mathematica, we can eliminate x between Eqs. (A1) and (A2), which results in
$$\begin{aligned} y^{5}+c_{1}y^{4}+c_{2}y^{3}+c_{3}y^{2}+c_{4}y+c_{5}=0, \end{aligned}$$
(A3)
where
$$\begin{aligned}&c_{1} = a_{1} b_{1}-a_{1}^{2}+2 a_{2}-5 b_{2}, \end{aligned}$$
(A4a)
$$\begin{aligned}&c_{2} = 4 a_{1}^{2} b_{2}-a_{2} a_{1} b_{1}-4 a_{1} b_{1} b_{2}+a_{2} b_{1}^{2}+3 a_{3} b_{1}-8 a_{2} b_{2}-2 a_{3} a_{1}+a_{2}^{2}+2 a_{4}+10 b_{2}^{2},\end{aligned}$$
(A4b)
$$\begin{aligned}&c_{3} = a_{3} b_{1}^{3}-a_{1} a_{3} b_{1}^{2}+4 a_{4} b_{1}^{2}-3 a_{2} b_{2} b_{1}^{2}+6 a_{1} b_{2}^{2} b_{1}+a_{2} a_{3} b_{1}-3 a_{1} a_{4} b_{1}+5 a_{5} b_{1}+3 a_{1} a_{2} b_{2} b_{1}-9 a_{3} b_{2} b_{1}\nonumber \\&-6 a_{1}^{2} b_{2}^{2}+12 a_{2} b_{2}^{2}-3 a_{2}^{2} b_{2}+6 a_{1} a_{3} b_{2}-6 a_{4} b_{2}-a_{3}^{2}+2 a_{2} a_{4}-2 a_{1} a_{5}-10 b_{2}^{3},\end{aligned}$$
(A4c)
$$\begin{aligned}&c_{4} = a_{4} b_{1}^{4}-a_{1} a_{4} b_{1}^{3}+5 a_{5} b_{1}^{3}-2 a_{3} b_{2} b_{1}^{3}+3 a_{2} b_{2}^{2} b_{1}^{2}+a_{2} a_{4} b_{1}^{2}-4 a_{1} a_{5} b_{1}^{2}+2 a_{1} a_{3} b_{2} b_{1}^{2}-8 a_{4} b_{2} b_{1}^{2}-4 a_{1} b_{2}^{3} b_{1}\nonumber \\&-3 a_{1} a_{2} b_{2}^{2} b_{1}+9 a_{3} b_{2}^{2} b_{1}-a_{3} a_{4} b_{1}+3 a_{2} a_{5} b_{1}-2 a_{2} a_{3} b_{2} b_{1}+6 a_{1} a_{4} b_{2} b_{1}-10 a_{5} b_{2} b_{1}\nonumber \\&+4 a_{1}^{2} b_{2}^{3}-8 a_{2} b_{2}^{3}+3 a_{2}^{2} b_{2}^{2}-6 a_{1} a_{3} b_{2}^{2}+6 a_{4} b_{2}^{2}+2 a_{3}^{2} b_{2}-4 a_{2} a_{4} b_{2}+4 a_{1} a_{5} b_{2}+a_{4}^{2}-2 a_{3} a_{5}+5 b_{2}^{4},\end{aligned}$$
(A4d)
$$\begin{aligned}&c_5 = a_5 b_{1}^{5}-a_{1} a_5 b_{1}^{4}+a_{2} a_5 b_{1}^{3}-a_{3} a_5 b_{1}^{2}+a_{4} a_{5} b_{1}-a_5^{2}. \end{aligned}$$
(A4e)
The two unknowns \(b_{1,2}\) allow for the elimination of \(c_{1,2}\). In fact, one can see that the equations \(c_{1}=c_{2}=0\) result in two quadratics, solving which provide the values
$$\begin{aligned}&b_{1}={\frac{4 a_{1}^{3}-13 a_{2} a_{1}\pm \sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}}{4 a_{1}^{2}-10 a_{2}}},\end{aligned}$$
(A5)
$$\begin{aligned}&b_{2} = {\frac{5 a_{2} a_{1}^{2}+\left( 15a_{3}\pm \sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}\right) a_{1}-20 a_{2}^{2}}{20 a_{1}^{2}-50 a_{2}}}. \end{aligned}$$
(A6)
Applying these values in the coefficients in Eq. (), the quintic (A3) reduces to the principal form
$$\begin{aligned} y^{5}+\mathfrak {u}y^{2}+\mathfrak {v}y+\mathfrak {w}=0, \end{aligned}$$
(A7)
in which
$$\begin{aligned} \mathfrak {u}= &\, {} {\frac{1}{40 \left( 2 a_{1}^{2}-5 a_{2}\right) ^{3}}}\left[ -90 \sqrt{5} a_{1} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}} a_{2}^{4}\right. \nonumber \\&\left. +48 \sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}} a_{1} \left( 2 a_{1}^{2}-5 a_{2}\right) a_{2}^{3}\right. \nonumber \\&\left. +1350 \sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}} a_{3} a_{2}^{3}\right. \nonumber \\&\left. -6 \sqrt{5} a_{1} \left( 2 a_{1}^{2}-5 a_{2}\right) ^{2} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}} a_{2}^{2}\right. \nonumber \\&\left. +320 \sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}} \left( 5 a_{2}-2 a_{1}^{2}\right) a_{3} a_{2}^{2}\right. \nonumber \\&\left. -2700 \sqrt{5} a_{1} a_{3}^{2} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}} a_{2}\right. \nonumber \\&\left. -46 \sqrt{5} \left( 2 a_{1}^{2}-5 a_{2}\right) ^{2} a_{3} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}} a_{2}\right. \nonumber \\&\left. +280 \sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}} a_{1} \left( 5 a_{2}-2 a_{1}^{2}\right) a_{4} a_{2}\right. \nonumber \\&\left. +4500 \sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}} a_{3}^{3}\right. \nonumber \\&\left. +520 \sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}} a_{1} \left( 2 a_{1}^{2}-5 a_{2}\right) a_{3}^{2}\right. \nonumber \\&\left. +8 \sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}} \left( 2 a_{1}^{2}-5 a_{2}\right) ^{3} a_{3}\right. \nonumber \\&\left. +675 \left( a_{2}^{6}-8 a_{1} a_{3} a_{2}^{4}+60 a_{3}^{2} a_{2}^{3}-80 a_{1} a_{3}^{3} a_{2}+100 a_{3}^{4}\right) +40 a_{1}^{2} \left( 2 a_{1}^{2}-5 a_{2}\right) ^{3} a_{4}\right. \nonumber \\&\left. +92 \sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}} a_{1} \left( 2 a_{1}^{2}-5 a_{2}\right) ^{2} a_{4}\right. \nonumber \\&\left. +1400 \sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}} \left( 2 a_{1}^{2}-5 a_{2}\right) a_{3} a_{4}\right. \nonumber \\&\left. +5 \left( 2 a_{1}^{2}-5 a_{2}\right) ^{3} \left( a_{2}^{3}-26 a_{4} a_{2}+44 a_{3}^{2}\right) +135 \left( 2 a_{1}^{2}-5 a_{2}\right) \left( -3 a_{2}^{5}+20 \left( a_{1} a_{3}+a_{4}\right) a_{2}^{3}-70 a_{3}^{2} a_{2}^{2}\right. \right. \nonumber \\&\left. \left. -80 a_{1} a_{3} a_{4} a_{2}+40 a_{3}^{2} \left( 2 a_{1} a_{3}+5 a_{4}\right) \right) \right. \nonumber \\&\left. +100 \sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}} \left( 2 a_{1}^{2}-5 a_{2}\right) ^{2} a_5\right. \nonumber \\&\left. -20 a_{1} \left( 2 a_{1}^{2}-5 a_{2}\right) ^{3} \left( a_{2} a_{3}-6 a_5\right) +5 \left( 2 a_{1}^{2}-5 a_{2}\right) ^{2} \left( 9 a_{2}^{4}-6 \left( 8 a_{1} a_{3}+33 a_{4}\right) a_{2}^{2}\right. \right. \nonumber \\&\left. \left. -2 \left( 137 a_{3}^{2}+30 a_{1} a_5\right) a_{2} +4 \left( 80 a_{4}^{2}+137 a_{1} a_{3} a_{4}+75 a_{3} a_5\right) \right) \right. \Big ], \end{aligned}$$
(A8)
$$\begin{aligned} \mathfrak {v}= &\, {}{\frac{a_{4} \left( 4 a_{1}^{3}-13 a_{2} a_{1}+\sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) ^{4}}{\left( 4 a_{1}^{2}-10 a_{2}\right)^{4}}}\nonumber \\&+{\frac{5 a_5 \left( 4 a_{1}^{3}-13 a_{2}a_{1}+\sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) ^{3}}{\left( 4 a_{1}^{2}-10 a_{2}\right) ^{3}}}\nonumber \\&-{\frac{a_{1} a_{4}\left( 4 a_{1}^{3}-13 a_{2} a_{1}+\sqrt{5} \sqrt{8 a_{3}a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2}a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) ^{3}}{\left( 4 a_{1}^{2}-10 a_{2}\right)^{3}}}\nonumber \\&+{\frac{a_{2} a_{4} \left( 4 a_{1}^{3}-13 a_{2}a_{1}+\sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) ^{2}}{\left( 4 a_{1}^{2}-10 a_{2}\right) ^{2}}}\nonumber \\&-{\frac{a_{1} a_5 \left( 4 a_{1}^{3}-13 a_{2} a_{1}+\sqrt{5} \sqrt{8 a_{3}a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2}a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) ^{2}}{\left( 2 a_{1}^{2}-5 a_{2}\right) ^{2}}}\nonumber \\&+{\frac{3 a_{2} a_5 \left( 4 a_{1}^{3}-13 a_{2} a_{1}+\sqrt{5}\sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right)a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2}a_{4}}+15 a_{3}\right) }{4 a_{1}^{2}-10 a_{2}}}\\&-{\frac{a_{3}a_{4} \left( 4 a_{1}^{3}-13 a_{2} a_{1}+\sqrt{5} \sqrt{8 a_{3}a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2}a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) }{4 a_{1}^{2}-10 a_{2}}}\nonumber \\&+{\frac{2 \left( 5 a_{2} a_{1}^{2}+\left( \sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) a_{1}-20 a_{2}^{2}\right) a_{3}^{2}}{20 a_{1}^{2}-50 a_{2}}}\nonumber \\&-{\frac{4 a_{2} a_{4} \left( 5 a_{2} a_{1}^{2}+\left( \sqrt{5}\sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right)a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2}a_{4}}+15 a_{3}\right) a_{1}-20 a_{2}^{2}\right) }{20 a_{1}^{2}-50 a_{2}}}\nonumber \\&-{\frac{a_{2} a_{3}}{10 \left( 2 a_{1}^{2}-5 a_{2}\right) ^{2}}}\left[ \left( 4 a_{1}^{3}-13 a_{2} a_{1}+\sqrt{5}\sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right)a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2}a_{4}}+15 a_{3}\right) \right. \nonumber \\&\left. \times \left( 5 a_{2} a_{1}^{2}+\left( \sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) a_{1}-20 a_{2}^{2}\right) \right] \nonumber \\&+{\frac{3 a_{1} a_{4}}{10 \left( 2 a_{1}^{2}-5 a_{2}\right) ^{2}}}\left[ \left( 4 a_{1}^{3}-13 a_{2} a_{1}+\sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) \right. \nonumber \\&\left. \times \left( 5 a_{2} a_{1}^{2}+\left( \sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) a_{1}-20 a_{2}^{2}\right) \right] \nonumber \\&+{\frac{a_{1} a_{3}}{20 \left( 2 a_{1}^{2}-5 a_{2}\right) ^{3}}}\left[ \left( 4 a_{1}^{3}-13 a_{2} a_{1}+\sqrt{5} \sqrt{8 a_{3}a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2}a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) ^{2}\right. \nonumber \\&\left. \times \left( 5 a_{2}a_{1}^{2}+\left( \sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) a_{1}-20 a_{2}^{2}\right)\right] \nonumber \\&-{\frac{a_{4}}{5 \left( 2 a_{1}^{2}-5 a_{2}\right) ^{3}}}\left[ \left( 4 a_{1}^{3}-13 a_{2} a_{1}+\sqrt{5}\sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right)a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2}a_{4}}+15 a_{3}\right) ^{2}\right. \nonumber \\&\left. \times \left(5 a_{2} a_{1}^{2}+\left( \sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) a_{1}-20 a_{2}^{2}\right) \right] \nonumber \\&-{\frac{a_{3}}{40 \left( 2 a_{1}^{2}-5 a_{2}\right) ^{4}}}\left[ \left( 4 a_{1}^{3}-13 a_{2}a_{1}+\sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) ^{3}\right. \nonumber \\&\left. \times \left( 5 a_{2} a_{1}^{2}+\left( \sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) a_{1}-20 a_{2}^{2}\right) \right] \nonumber \\&+{\frac{3 a_{2}^{2} \left( 5 a_{2} a_{1}^{2}+\left( \sqrt{5}\sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right)a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2}a_{4}}+15 a_{3}\right) a_{1}-20 a_{2}^{2}\right) ^{2}}{\left( 20 a_{1}^{2}-50 a_{2}\right) ^{2}}}\nonumber \\&-{\frac{6 a_{1} a_{3}\left( 5 a_{2} a_{1}^{2}+\left( \sqrt{5} \sqrt{8 a_{3}a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2}a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) a_{1}-20 a_{2}^{2}\right) ^{2}}{\left( 20 a_{1}^{2}-50 a_{2}\right) ^{2}}}\nonumber \\&+{\frac{6 a_{4} \left( 5 a_{2}a_{1}^{2}+\left( \sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) a_{1}-20 a_{2}^{2}\right)^{2}}{\left( 20 a_{1}^{2}-50 a_{2}\right) ^{2}}}\nonumber \\&-{\frac{3 a_{1} a_{2}}{200 \left( 2 a_{1}^{2}-5 a_{2}\right)^{3}}}\left[ \left( 4 a_{1}^{3}-13 a_{2} a_{1}+\sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) \right. \nonumber \\& \left. \times \left( 5 a_{2} a_{1}^{2}+\left( \sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) a_{1}-20 a_{2}^{2}\right) ^{2}\right] \nonumber \\& +{\frac{9 a_{3}}{200 \left( 2 a_{1}^{2}-5 a_{2}\right) ^{3}}} \left[ \left( 4 a_{1}^{3}-13 a_{2} a_{1}+\sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) \right.\nonumber \\& \left. \times \left( 5 a_{2} a_{1}^{2}+\left(\sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right)a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2}a_{4}}+15 a_{3}\right) a_{1}-20 a_{2}^{2}\right) ^{2}\right]\nonumber \\& +{\frac{3 a_{2}}{400 \left( 2 a_{1}^{2}-5 a_{2}\right) ^{4}}}\left[ \left( 4 a_{1}^{3}-13 a_{2} a_{1}+\sqrt{5}\sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right)a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2}a_{4}}+15 a_{3}\right) ^{2}\right. \nonumber \\& \left. \times \left( 5 a_{2} a_{1}^{2}+\left( \sqrt{5} \sqrt{8 a_{3}a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2}a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) a_{1}-20 a_{2}^{2}\right) ^{2}\right] \nonumber \\& + \frac{{4a_{1}^{2} \left( {5a_{2} a_{1}^{2} + \left({\sqrt 5 \sqrt {8a_{3} a_{1}^{3} + \left( {16a_{4} - 3a_{2}^{2} }\right)a_{1}^{2} - 38a_{2} a_{3} a_{1} + 12a_{2}^{3} +45a_{3}^{2} - 40a_{2} a_{4} } + 15a_{3} } \right)a_{1} -20a_{2}^{2} } \right)^{3} }}{{\left( {20a_{1}^{2} - 50a_{2} ^{3} }\right)}} \\& - {\frac{{8a_{2} \left( {5a_{2} a_{1}^{2} +\left( {\sqrt 5 \sqrt {8a_{3} a_{1}^{3} + \left( {16a_{4} -3a_{2}^{2} } \right)a_{1}^{2} - 38a_{2} a_{3} a_{1} + 12a_{2}^{3}+ 45a_{3}^{2} - 40a_{2} a_{4} } + 15a_{3} } \right)a_{1} -20a_{2}^{2} } \right)^{3} }}{{\left( {20a_{1}^{2} - 50a_{2} }\right)^{3} }}} \nonumber \\& -{\frac{a_{1}}{500 \left( 2 a_{1}^{2}-5 a_{2}\right) ^{4}}}\left[ \left( 4 a_{1}^{3}-13 a_{2}a_{1}+\sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) \right.\end{aligned}$$
$$\begin{aligned}& \left. \qquad\times \left( 5 a_{2} a_{1}^{2}+\left( \sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) a_{1}-20 a_{2}^{2}\right) ^{3}\right] \nonumber \\&\qquad +{\frac{5 \left( 5 a_{2} a_{1}^{2}+\left( \sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) a_{1}-20 a_{2}^{2}\right) ^{4}}{\left( 20 a_{1}^{2}-50 a_{2}\right)^{4}}}\nonumber \\&\qquad +{\frac{3 a_{2} \left( 4 a_{1}^{3}-13 a_{2} a_{1}+\sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) a_5}{4 a_{1}^{2}-10 a_{2}}}-2 a_{3} a_5\nonumber \\&\qquad -{\frac{a_{1} \left( 4 a_{1}^{3}-13 a_{2} a_{1}+\sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) ^{2}a_5}{\left( 2 a_{1}^{2}-5 a_{2}\right) ^{2}}}\nonumber \\&\qquad +{\frac{5 \left( 4 a_{1}^{3}-13 a_{2} a_{1}+\sqrt{5} \sqrt{8 a_{3}a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2}a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) ^{3} a_5}{\left( 4 a_{1}^{2}-10 a_{2}\right)^{3}}}\nonumber \\&\qquad +{\frac{4 a_{1} \left( 5 a_{2}a_{1}^{2}+\left( \sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) a_{1}-20 a_{2}^{2}\right)a_5}{20 a_{1}^{2}-50 a_{2}}}\nonumber \\&\qquad -{\frac{a_5}{2 \left( 2 a_{1}^{2}-5 a_{2}\right) ^{2}}}\left[ \left( 4 a_{1}^{3}-13 a_{2} a_{1}+\sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) \right. \nonumber \\&\qquad \left. \times \left( 5 a_{2} a_{1}^{2}+\left( \sqrt{5}\sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right)a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2}a_{4}}+15 a_{3}\right) a_{1}-20 a_{2}^{2}\right) \right] ,\end{aligned}$$
(A9)
$$\begin{aligned} \mathfrak {w}= &\, {}{\frac{a_{4} \left( 4 a_{1}^{3}-13 a_{2} a_{1}+\sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) a_5}{4 a_{1}^{2}-10 a_{2}}}-a_5^{2}\nonumber \\&-{\frac{a_{3} \left( 4 a_{1}^{3}-13 a_{2} a_{1}+\sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) ^{2} a_5}{\left( 4 a_{1}^{2}-10 a_{2}\right)^{2}}}\nonumber \\&+{\frac{a_{2} \left( 4 a_{1}^{3}-13 a_{2}a_{1}+\sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) ^{3} a_5}{\left( 4 a_{1}^{2}-10 a_{2}\right) ^{3}}}\nonumber \\&-{\frac{a_{1} \left( 4 a_{1}^{3}-13 a_{2} a_{1}+\sqrt{5} \sqrt{8 a_{3} a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2} a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) ^{4}a_5}{\left( 4 a_{1}^{2}-10 a_{2}\right) ^{4}}}\nonumber \\&+{\frac{\left( 4 a_{1}^{3}-13 a_{2} a_{1}+\sqrt{5} \sqrt{8 a_{3}a_{1}^{3}+\left( 16 a_{4}-3 a_{2}^{2}\right) a_{1}^{2}-38 a_{2}a_{3} a_{1}+12 a_{2}^{3}+45 a_{3}^{2}-40 a_{2} a_{4}}+15 a_{3}\right) ^{5} a_5}{\left( 4 a_{1}^{2}-10 a_{2}\right) ^{5}}}.\end{aligned}$$
(A10)
Now, to transform the principal quintic (A7) to its Bring–Jerrard form, we use the quartic Tschirnhausen transformation
$$\begin{aligned} z = y^{4}+\mathfrak {p}y^{3}+\mathfrak {q}y^{2}+\mathfrak {r}y+ \mathfrak {s}.\end{aligned}$$
(A11)
Eliminating y between Eqs. (A7) and (A11), we get to the quintic
$$\begin{aligned}z^{5}+d_{1}z^{4}+d_{2}z^{3}+d_{3}z^{2}+d_{4}z+d_5=0,\end{aligned}$$
(A12)
in which
$$\begin{aligned}&d_{1} = 3 \mathfrak {p}\mathfrak {u}-5 \mathfrak {s}+4 \mathfrak {v},\end{aligned}$$
(A13a)
$$\begin{aligned} d_{2} =& 10 \mathfrak {s}^{2} - 12 \mathfrak {p}\mathfrak {s}\mathfrak {u}+ 3 \mathfrak {p}^{2} \mathfrak {u}^{2} - 3 \mathfrak {q}\mathfrak {u}^{2} + 2 \mathfrak {q}^{2} \mathfrak {v}- 16 \mathfrak {s}\mathfrak {v}+ 5 \mathfrak {p}\mathfrak {u}\mathfrak {v}+ 6 \mathfrak {v}^{2} + 5 \mathfrak {p}\mathfrak {q}\mathfrak {w}- 4 \mathfrak {u}\mathfrak {w}+ \mathfrak {r}(3 \mathfrak {q}\mathfrak {u}+ 4 \mathfrak {p}\mathfrak {v}+ 5 \mathfrak {w}),\end{aligned}$$
(A13b)
$$\begin{aligned}d_{3} =&\, 7 \mathfrak {p}^{2} \mathfrak {q}\mathfrak {u}\mathfrak {w}-4 \mathfrak {p}^{2}\mathfrak {q}\mathfrak {v}^{2}+5 \mathfrak {p}^{2} \mathfrak {r}\mathfrak {u}\mathfrak {v}-9 \mathfrak {p}^{2} \mathfrak {s}\mathfrak {u}^{2}+\mathfrak {p}^{2} \mathfrak {u}^{2} \mathfrak {v}+\mathfrak {p}^{3} \mathfrak {u}^{3}-3 \mathfrak {p}^{3}\mathfrak {v}\mathfrak {w}-5 \mathfrak {p}^{2} \mathfrak {w}^{2}-\mathfrak {p}\mathfrak {q}^{2} \mathfrak {u}\mathfrak {v}+3 \mathfrak {p}\mathfrak {q}\mathfrak {r}\mathfrak {u}^{2}-15 \mathfrak {p}\mathfrak {q}\mathfrak {s}\mathfrak {w}\nonumber \\&\qquad -3 \mathfrak {p}\mathfrak {q}\mathfrak {u}^{3}+2 \mathfrak {p}\mathfrak {q}\mathfrak {v}\mathfrak {w}+5 \mathfrak {p}\mathfrak {r}^{2} \mathfrak {w}-12 \mathfrak {p}\mathfrak {r}\mathfrak {s}\mathfrak {v}-\mathfrak {p}\mathfrak {r}\mathfrak {u}\mathfrak {w}+8 \mathfrak {p}\mathfrak {r}\mathfrak {v}^{2}+18 \mathfrak {p}\mathfrak {s}^{2} \mathfrak {u}-15 \mathfrak {p}\mathfrak {s}\mathfrak {u}\mathfrak {v}-\mathfrak {p}\mathfrak {u}^{2} \mathfrak {w}+\mathfrak {p}\mathfrak {u}\mathfrak {v}^{2}+5 \mathfrak {q}^{2} \mathfrak {r}\mathfrak {w}\nonumber \\&\qquad -6 \mathfrak {q}^{2} \mathfrak {s}\mathfrak {v}-\mathfrak {q}^{3} \mathfrak {u}^{2}-8 \mathfrak {q}^{2} \mathfrak {u}\mathfrak {w}+4 \mathfrak {q}^{2} \mathfrak {v}^{2}+4 \mathfrak {q}\mathfrak {r}^{2} \mathfrak {v}-9 \mathfrak {q}\mathfrak {r}\mathfrak {s}\mathfrak {u}-2 \mathfrak {q}\mathfrak {r}\mathfrak {u}\mathfrak {v}+9 \mathfrak {q}\mathfrak {s}\mathfrak {u}^{2}-2 \mathfrak {q}\mathfrak {u}^{2} \mathfrak {v}-5 \mathfrak {q}\mathfrak {w}^{2}-3 \mathfrak {r}^{2} \mathfrak {u}^{2}+\mathfrak {r}^{3} \mathfrak {u}\nonumber \\&\qquad -15 \mathfrak {r}\mathfrak {s}\mathfrak {w}+3 \mathfrak {r}\mathfrak {u}^{3}+11 \mathfrak {r}\mathfrak {v}\mathfrak {w}+24 \mathfrak {s}^{2} \mathfrak {v}-10 \mathfrak {s}^{3}+12 \mathfrak {s}\mathfrak {u}\mathfrak {w}-18 \mathfrak {s}\mathfrak {v}^{2}-\mathfrak {u}^{4}-8 \mathfrak {u}\mathfrak {v}\mathfrak {w}+4 \mathfrak {v}^{3},\end{aligned}$$
(A13c)
$$\begin{aligned}d_{4} = &\,5 \mathfrak {s}^{4} - 2 \mathfrak {r}^{3} \mathfrak {s}\mathfrak {u}+ 9 \mathfrak {q}\mathfrak {r}\mathfrak {s}^{2} \mathfrak {u}- 12 \mathfrak {p}\mathfrak {s}^{3} \mathfrak {u}+ 2 \mathfrak {q}^{3} \mathfrak {s}\mathfrak {u}^{2} - 6 \mathfrak {p}\mathfrak {q}\mathfrak {r}\mathfrak {s}\mathfrak {u}^{2} + 6 \mathfrak {r}^{2} \mathfrak {s}\mathfrak {u}^{2} + 9 \mathfrak {p}^{2} \mathfrak {s}^{2} \mathfrak {u}^{2} - 9 \mathfrak {q}\mathfrak {s}^{2} \mathfrak {u}^{2} - 2 \mathfrak {p}^{3} \mathfrak {s}\mathfrak {u}^{3} + 6 \mathfrak {p}\mathfrak {q}\mathfrak {s}\mathfrak {u}^{3} \nonumber \\&\qquad - 6 \mathfrak {r}\mathfrak {s}\mathfrak {u}^{3} + 2 \mathfrak {s}\mathfrak {u}^{4} + \mathfrak {r}^{4} \mathfrak {v}- 8 \mathfrak {q}\mathfrak {r}^{2} \mathfrak {s}\mathfrak {v}+ 6 \mathfrak {q}^{2} \mathfrak {s}^{2} \mathfrak {v}+ 12 \mathfrak {p}\mathfrak {r}\mathfrak {s}^{2} \mathfrak {v}- 16 \mathfrak {s}^{3} \mathfrak {v}- \mathfrak {q}^{3} \mathfrak {r}\mathfrak {u}\mathfrak {v}+ 3 \mathfrak {p}\mathfrak {q}\mathfrak {r}^{2} \mathfrak {u}\mathfrak {v}- 3 \mathfrak {r}^{3} \mathfrak {u}\mathfrak {v}+ 2 \mathfrak {p}\mathfrak {q}^{2} \mathfrak {s}\mathfrak {u}\mathfrak {v}\nonumber \\&\qquad - 10 \mathfrak {p}^{2} \mathfrak {r}\mathfrak {s}\mathfrak {u}\mathfrak {v}+ 4 \mathfrak {q}\mathfrak {r}\mathfrak {s}\mathfrak {u}\mathfrak {v}+ 15 \mathfrak {p}\mathfrak {s}^{2} \mathfrak {u}\mathfrak {v}+ \mathfrak {p}^{3} \mathfrak {r}\mathfrak {u}^{2} \mathfrak {v}- 3 \mathfrak {p}\mathfrak {q}\mathfrak {r}\mathfrak {u}^{2} \mathfrak {v}+ 3 \mathfrak {r}^{2} \mathfrak {u}^{2} \mathfrak {v}- 2 \mathfrak {p}^{2} \mathfrak {s}\mathfrak {u}^{2} \mathfrak {v}+ 4 \mathfrak {q}\mathfrak {s}\mathfrak {u}^{2} \mathfrak {v}- \mathfrak {r}\mathfrak {u}^{3} \mathfrak {v}+ \mathfrak {q}^{4} \mathfrak {v}^{2} \nonumber \\&\qquad - 4 \mathfrak {p}\mathfrak {q}^{2} \mathfrak {r}\mathfrak {v}^{2} + 2 \mathfrak {p}^{2} \mathfrak {r}^{2} \mathfrak {v}^{2} + 4 \mathfrak {q}\mathfrak {r}^{2} \mathfrak {v}^{2} + 8 \mathfrak {p}^{2} \mathfrak {q}\mathfrak {s}\mathfrak {v}^{2} - 8 \mathfrak {q}^{2} \mathfrak {s}\mathfrak {v}^{2} - 16 \mathfrak {p}\mathfrak {r}\mathfrak {s}\mathfrak {v}^{2} + 18 \mathfrak {s}^{2} \mathfrak {v}^{2} - \mathfrak {p}^{3} \mathfrak {q}\mathfrak {u}\mathfrak {v}^{2} + 3 \mathfrak {p}\mathfrak {q}^{2} \mathfrak {u}\mathfrak {v}^{2} + \mathfrak {p}^{2} \mathfrak {r}\mathfrak {u}\mathfrak {v}^{2} \nonumber \\&\qquad - 5 \mathfrak {q}\mathfrak {r}\mathfrak {u}\mathfrak {v}^{2} - 2 \mathfrak {p}\mathfrak {s}\mathfrak {u}\mathfrak {v}^{2} + \mathfrak {q}\mathfrak {u}^{2} \mathfrak {v}^{2} + \mathfrak {p}^{4} \mathfrak {v}^{3} - 4 \mathfrak {p}^{2} \mathfrak {q}\mathfrak {v}^{3} + 2 \mathfrak {q}^{2} \mathfrak {v}^{3} + 4 \mathfrak {p}\mathfrak {r}\mathfrak {v}^{3} - 8 \mathfrak {s}\mathfrak {v}^{3} - \mathfrak {p}\mathfrak {u}\mathfrak {v}^{3} + \mathfrak {v}^{4} + 5 \mathfrak {q}\mathfrak {r}^{3} \mathfrak {w}- 10 \mathfrak {q}^{2} \mathfrak {r}\mathfrak {s}\mathfrak {w}\nonumber \\&\qquad - 10 \mathfrak {p}\mathfrak {r}^{2} \mathfrak {s}\mathfrak {w}+ 15 \mathfrak {p}\mathfrak {q}\mathfrak {s}^{2} \mathfrak {w}+ 15 \mathfrak {r}\mathfrak {s}^{2} \mathfrak {w}- 2 \mathfrak {q}^{4} \mathfrak {u}\mathfrak {w}+ 6 \mathfrak {p}\mathfrak {q}^{2} \mathfrak {r}\mathfrak {u}\mathfrak {w}+ 3 \mathfrak {p}^{2} \mathfrak {r}^{2} \mathfrak {u}\mathfrak {w}- 9 \mathfrak {q}\mathfrak {r}^{2} \mathfrak {u}\mathfrak {w}- 14 \mathfrak {p}^{2} \mathfrak {q}\mathfrak {s}\mathfrak {u}\mathfrak {w}+ 16 \mathfrak {q}^{2} \mathfrak {s}\mathfrak {u}\mathfrak {w}\nonumber \\&\qquad +2 \mathfrak {p}\mathfrak {r}\mathfrak {s}\mathfrak {u}\mathfrak {w}- 12 \mathfrak {s}^{2} \mathfrak {u}\mathfrak {w}+ 2 \mathfrak {p}^{3} \mathfrak {q}\mathfrak {u}^{2} \mathfrak {w}- 6 \mathfrak {p}\mathfrak {q}^{2} \mathfrak {u}^{2} \mathfrak {w}+ 6 \mathfrak {q}\mathfrak {r}\mathfrak {u}^{2} \mathfrak {w}+ 2 \mathfrak {p}\mathfrak {s}\mathfrak {u}^{2} \mathfrak {w}- 2 \mathfrak {q}\mathfrak {u}^{3} \mathfrak {w}+ \mathfrak {p}\mathfrak {q}^{3} \mathfrak {v}\mathfrak {w}- 7 \mathfrak {p}^{2} \mathfrak {q}\mathfrak {r}\mathfrak {v}\mathfrak {w}+ 3 \mathfrak {q}^{2} \mathfrak {r}\mathfrak {v}\mathfrak {w}\nonumber \\&\qquad + 13 \mathfrak {p}\mathfrak {r}^{2} \mathfrak {v}\mathfrak {w}+ 6 \mathfrak {p}^{3} \mathfrak {s}\mathfrak {v}\mathfrak {w}- 4 \mathfrak {p}\mathfrak {q}\mathfrak {s}\mathfrak {v}\mathfrak {w}- 22 \mathfrak {r}\mathfrak {s}\mathfrak {v}\mathfrak {w}- 3 \mathfrak {p}^{4} \mathfrak {u}\mathfrak {v}\mathfrak {w}+ 11 \mathfrak {p}^{2} \mathfrak {q}\mathfrak {u}\mathfrak {v}\mathfrak {w}- 4 \mathfrak {q}^{2} \mathfrak {u}\mathfrak {v}\mathfrak {w}- 10 \mathfrak {p}\mathfrak {r}\mathfrak {u}\mathfrak {v}\mathfrak {w}+ 16 \mathfrak {s}\mathfrak {u}\mathfrak {v}\mathfrak {w}\nonumber \\&\qquad + 3 \mathfrak {p}\mathfrak {u}^{2} \mathfrak {v}\mathfrak {w}+ \mathfrak {p}^{3} \mathfrak {v}^{2} \mathfrak {w}- 3 \mathfrak {p}\mathfrak {q}\mathfrak {v}^{2} \mathfrak {w}+ 7 \mathfrak {r}\mathfrak {v}^{2} \mathfrak {w}- 4 \mathfrak {u}\mathfrak {v}^{2} \mathfrak {w}+ 5 \mathfrak {p}^{2} \mathfrak {q}^{2} \mathfrak {w}^{2} - 5 \mathfrak {q}^{3} \mathfrak {w}^{2} - 5 \mathfrak {p}^{3} \mathfrak {r}\mathfrak {w}^{2} - 5 \mathfrak {p}\mathfrak {q}\mathfrak {r}\mathfrak {w}^{2} + 5 \mathfrak {r}^{2} \mathfrak {w}^{2} \nonumber \\&\qquad + 10 \mathfrak {p}^{2} \mathfrak {s}\mathfrak {w}^{2} + 10 \mathfrak {q}\mathfrak {s}\mathfrak {w}^{2} - 2 \mathfrak {p}^{3} \mathfrak {u}\mathfrak {w}^{2} + 4 \mathfrak {p}\mathfrak {q}\mathfrak {u}\mathfrak {w}^{2} - 7 \mathfrak {r}\mathfrak {u}\mathfrak {w}^{2} + 2 \mathfrak {u}^{2} \mathfrak {w}^{2} + \mathfrak {p}^{2} \mathfrak {v}\mathfrak {w}^{2} - 6 \mathfrak {q}\mathfrak {v}\mathfrak {w}^{2} + 5 \mathfrak {p}\mathfrak {w}^{3},\end{aligned}$$
(A13d)
$$\begin{aligned}d_5 =&\, \mathfrak {w}^{3} \mathfrak {p}^{5}-\mathfrak {s}\mathfrak {v}^{3} \mathfrak {p}^{4}-2 \mathfrak {r}\mathfrak {u}\mathfrak {w}^{2} \mathfrak {p}^{4}-\mathfrak {q}\mathfrak {v}\mathfrak {w}^{2} \mathfrak {p}^{4}+\mathfrak {r}\mathfrak {v}^{2} \mathfrak {w}\mathfrak {p}^{4}+3 \mathfrak {s}\mathfrak {u}\mathfrak {v}\mathfrak {w}\mathfrak {p}^{4}+\mathfrak {s}^{2} \mathfrak {u}^{3} \mathfrak {p}^{3}-5 \mathfrak {q}\mathfrak {w}^{3} \mathfrak {p}^{3}+\mathfrak {q}\mathfrak {s}\mathfrak {u}\mathfrak {v}^{2} \mathfrak {p}^{3}+5 \mathfrak {r}\mathfrak {s}\mathfrak {w}^{2} \mathfrak {p}^{3}\nonumber \\&\qquad +\mathfrak {q}^{2} \mathfrak {u}\mathfrak {w}^{2} \mathfrak {p}^{3}+2 \mathfrak {s}\mathfrak {u}\mathfrak {w}^{2} \mathfrak {p}^{3}+\mathfrak {r}\mathfrak {v}\mathfrak {w}^{2} \mathfrak {p}^{3}-\mathfrak {r}\mathfrak {s}\mathfrak {u}^{2} \mathfrak {v}\mathfrak {p}^{3}+\mathfrak {r}^{2} \mathfrak {u}^{2} \mathfrak {w}\mathfrak {p}^{3}-2 \mathfrak {q}\mathfrak {s}\mathfrak {u}^{2} \mathfrak {w}\mathfrak {p}^{3}-\mathfrak {s}\mathfrak {v}^{2} \mathfrak {w}\mathfrak {p}^{3}-3 \mathfrak {s}^{2} \mathfrak {v}\mathfrak {w}\mathfrak {p}^{3}-\mathfrak {q}\mathfrak {r}\mathfrak {u}\mathfrak {v}\mathfrak {w}\mathfrak {p}^{3}\nonumber \\&\qquad +4 \mathfrak {q}\mathfrak {s}\mathfrak {v}^{3} \mathfrak {p}^{2}+5 \mathfrak {r}\mathfrak {w}^{3} \mathfrak {p}^{2}-\mathfrak {u}\mathfrak {w}^{3} \mathfrak {p}^{2}-3 \mathfrak {s}^{3} \mathfrak {u}^{2} \mathfrak {p}^{2}-4 \mathfrak {q}\mathfrak {s}^{2} \mathfrak {v}^{2} \mathfrak {p}^{2}-2 \mathfrak {r}^{2} \mathfrak {s}\mathfrak {v}^{2} \mathfrak {p}^{2}-\mathfrak {r}\mathfrak {s}\mathfrak {u}\mathfrak {v}^{2} \mathfrak {p}^{2}-5 \mathfrak {q}\mathfrak {r}^{2} \mathfrak {w}^{2} \mathfrak {p}^{2}-5 \mathfrak {s}^{2} \mathfrak {w}^{2} \mathfrak {p}^{2}\nonumber \\&\qquad -5 \mathfrak {q}^{2} \mathfrak {s}\mathfrak {w}^{2} \mathfrak {p}^{2}+6 \mathfrak {q}\mathfrak {r}\mathfrak {u}\mathfrak {w}^{2} \mathfrak {p}^{2}+4 \mathfrak {q}^{2} \mathfrak {v}\mathfrak {w}^{2} \mathfrak {p}^{2}-\mathfrak {s}\mathfrak {v}\mathfrak {w}^{2} \mathfrak {p}^{2}+\mathfrak {s}^{2} \mathfrak {u}^{2} \mathfrak {v}\mathfrak {p}^{2}+5 \mathfrak {r}\mathfrak {s}^{2} \mathfrak {u}\mathfrak {v}\mathfrak {p}^{2}-4 \mathfrak {q}\mathfrak {r}\mathfrak {v}^{2} \mathfrak {w}\mathfrak {p}^{2}+7 \mathfrak {q}\mathfrak {s}^{2} \mathfrak {u}\mathfrak {w}\mathfrak {p}^{2}-3 \mathfrak {r}^{2} \mathfrak {s}\mathfrak {u}\mathfrak {w}\mathfrak {p}^{2}\nonumber \\&\qquad +2 \mathfrak {r}^{3} \mathfrak {v}\mathfrak {w}\mathfrak {p}^{2}+7 \mathfrak {q}\mathfrak {r}\mathfrak {s}\mathfrak {v}\mathfrak {w}\mathfrak {p}^{2}+\mathfrak {r}^{2} \mathfrak {u}\mathfrak {v}\mathfrak {w}\mathfrak {p}^{2}-11 \mathfrak {q}\mathfrak {s}\mathfrak {u}\mathfrak {v}\mathfrak {w}\mathfrak {p}^{2}-3 \mathfrak {q}\mathfrak {s}^{2} \mathfrak {u}^{3} \mathfrak {p}-4 \mathfrak {r}\mathfrak {s}\mathfrak {v}^{3} \mathfrak {p}+\mathfrak {s}\mathfrak {u}\mathfrak {v}^{3} \mathfrak {p}+5 \mathfrak {q}^{2} \mathfrak {w}^{3} \mathfrak {p}-5 \mathfrak {s}\mathfrak {w}^{3} \mathfrak {p}+\mathfrak {v}\mathfrak {w}^{3} \mathfrak {p}\nonumber \\&\qquad +3 \mathfrak {q}\mathfrak {r}\mathfrak {s}^{2} \mathfrak {u}^{2} \mathfrak {p}+8 \mathfrak {r}\mathfrak {s}^{2} \mathfrak {v}^{2} \mathfrak {p}+4 \mathfrak {q}^{2} \mathfrak {r}\mathfrak {s}\mathfrak {v}^{2} \mathfrak {p}+\mathfrak {s}^{2} \mathfrak {u}\mathfrak {v}^{2} \mathfrak {p}-3 \mathfrak {q}^{2} \mathfrak {s}\mathfrak {u}\mathfrak {v}^{2} \mathfrak {p}+5 \mathfrak {r}^{3} \mathfrak {w}^{2} \mathfrak {p}+2 \mathfrak {r}\mathfrak {u}^{2} \mathfrak {w}^{2} \mathfrak {p}+5 \mathfrak {q}^{3} \mathfrak {r}\mathfrak {w}^{2} \mathfrak {p}+5 \mathfrak {q}\mathfrak {r}\mathfrak {s}\mathfrak {w}^{2} \mathfrak {p}\nonumber \\&\qquad -3 \mathfrak {q}^{3} \mathfrak {u}\mathfrak {w}^{2} \mathfrak {p}-7 \mathfrak {r}^{2} \mathfrak {u}\mathfrak {w}^{2} \mathfrak {p}-4 \mathfrak {q}\mathfrak {s}\mathfrak {u}\mathfrak {w}^{2} \mathfrak {p}-7 \mathfrak {q}\mathfrak {r}\mathfrak {v}\mathfrak {w}^{2} \mathfrak {p}+\mathfrak {q}\mathfrak {u}\mathfrak {v}\mathfrak {w}^{2} \mathfrak {p}+3 \mathfrak {s}^{4} \mathfrak {u}\mathfrak {p}-4 \mathfrak {r}\mathfrak {s}^{3} \mathfrak {v}\mathfrak {p}+3 \mathfrak {q}\mathfrak {r}\mathfrak {s}\mathfrak {u}^{2} \mathfrak {v}\mathfrak {p}-5 \mathfrak {s}^{3} \mathfrak {u}\mathfrak {v}\mathfrak {p}-\mathfrak {q}^{2} \mathfrak {s}^{2} \mathfrak {u}\mathfrak {v}\mathfrak {p}\nonumber \\&\qquad -3 \mathfrak {q}\mathfrak {r}^{2} \mathfrak {s}\mathfrak {u}\mathfrak {v}\mathfrak {p}-5 \mathfrak {q}\mathfrak {s}^{3} \mathfrak {w}\mathfrak {p}+5 \mathfrak {r}^{2} \mathfrak {s}^{2} \mathfrak {w}\mathfrak {p}-3 \mathfrak {q}\mathfrak {r}^{2} \mathfrak {u}^{2} \mathfrak {w}\mathfrak {p}-\mathfrak {s}^{2} \mathfrak {u}^{2} \mathfrak {w}\mathfrak {p}+6 \mathfrak {q}^{2} \mathfrak {s}\mathfrak {u}^{2} \mathfrak {w}\mathfrak {p}+4 \mathfrak {r}^{2} \mathfrak {v}^{2} \mathfrak {w}\mathfrak {p}+3 \mathfrak {q}\mathfrak {s}\mathfrak {v}^{2} \mathfrak {w}\mathfrak {p}-\mathfrak {r}\mathfrak {u}\mathfrak {v}^{2} \mathfrak {w}\mathfrak {p}\nonumber \\&\qquad +3 \mathfrak {q}\mathfrak {r}^{3} \mathfrak {u}\mathfrak {w}\mathfrak {p}-\mathfrak {r}\mathfrak {s}^{2} \mathfrak {u}\mathfrak {w}\mathfrak {p}-6 \mathfrak {q}^{2} \mathfrak {r}\mathfrak {s}\mathfrak {u}\mathfrak {w}\mathfrak {p}-4 \mathfrak {q}^{2} \mathfrak {r}^{2} \mathfrak {v}\mathfrak {w}\mathfrak {p}+2 \mathfrak {q}\mathfrak {s}^{2} \mathfrak {v}\mathfrak {w}\mathfrak {p}-3 \mathfrak {s}\mathfrak {u}^{2} \mathfrak {v}\mathfrak {w}\mathfrak {p}-\mathfrak {q}^{3} \mathfrak {s}\mathfrak {v}\mathfrak {w}\mathfrak {p}-13 \mathfrak {r}^{2} \mathfrak {s}\mathfrak {v}\mathfrak {w}\mathfrak {p}+3 \mathfrak {q}^{2} \mathfrak {r}\mathfrak {u}\mathfrak {v}\mathfrak {w}\mathfrak {p}\nonumber \\&\qquad +10 \mathfrak {r}\mathfrak {s}\mathfrak {u}\mathfrak {v}\mathfrak {w}\mathfrak {p}-\mathfrak {s}^{5}-\mathfrak {s}^{2} \mathfrak {u}^{4}-\mathfrak {s}\mathfrak {v}^{4}-\mathfrak {w}^{4}+3 \mathfrak {r}\mathfrak {s}^{2} \mathfrak {u}^{3}+4 \mathfrak {s}^{2} \mathfrak {v}^{3}-2 \mathfrak {q}^{2} \mathfrak {s}\mathfrak {v}^{3}-5 \mathfrak {q}\mathfrak {r}\mathfrak {w}^{3}+2 \mathfrak {q}\mathfrak {u}\mathfrak {w}^{3}+3 \mathfrak {q}\mathfrak {s}^{3} \mathfrak {u}^{2}-\mathfrak {q}^{3} \mathfrak {s}^{2} \mathfrak {u}^{2}\nonumber \\&\qquad -3 \mathfrak {r}^{2} \mathfrak {s}^{2} \mathfrak {u}^{2}-6 \mathfrak {s}^{3} \mathfrak {v}^{2}+4 \mathfrak {q}^{2} \mathfrak {s}^{2} \mathfrak {v}^{2}-\mathfrak {q}\mathfrak {s}\mathfrak {u}^{2} \mathfrak {v}^{2}-\mathfrak {q}^{4} \mathfrak {s}\mathfrak {v}^{2}-4 \mathfrak {q}\mathfrak {r}^{2} \mathfrak {s}\mathfrak {v}^{2}+5 \mathfrak {q}\mathfrak {r}\mathfrak {s}\mathfrak {u}\mathfrak {v}^{2}-\mathfrak {q}^{5} \mathfrak {w}^{2}-5 \mathfrak {q}^{2} \mathfrak {r}^{2} \mathfrak {w}^{2}-5 \mathfrak {q}\mathfrak {s}^{2} \mathfrak {w}^{2}-\mathfrak {q}^{2} \mathfrak {u}^{2} \mathfrak {w}^{2}\nonumber \\&\qquad -2 \mathfrak {s}\mathfrak {u}^{2} \mathfrak {w}^{2}-\mathfrak {q}\mathfrak {v}^{2} \mathfrak {w}^{2}+5 \mathfrak {q}^{3} \mathfrak {s}\mathfrak {w}^{2}-5 \mathfrak {r}^{2} \mathfrak {s}\mathfrak {w}^{2}+3 \mathfrak {q}^{2} \mathfrak {r}\mathfrak {u}\mathfrak {w}^{2}+7 \mathfrak {r}\mathfrak {s}\mathfrak {u}\mathfrak {w}^{2}-2 \mathfrak {q}^{3} \mathfrak {v}\mathfrak {w}^{2}+3 \mathfrak {r}^{2} \mathfrak {v}\mathfrak {w}^{2}+6 \mathfrak {q}\mathfrak {s}\mathfrak {v}\mathfrak {w}^{2}-3 \mathfrak {r}\mathfrak {u}\mathfrak {v}\mathfrak {w}^{2}\nonumber \\&\qquad -3 \mathfrak {q}\mathfrak {r}\mathfrak {s}^{3} \mathfrak {u}+\mathfrak {r}^{3} \mathfrak {s}^{2} \mathfrak {u}+4 \mathfrak {s}^{4} \mathfrak {v}-2 \mathfrak {q}^{2} \mathfrak {s}^{3} \mathfrak {v}+\mathfrak {r}\mathfrak {s}\mathfrak {u}^{3} \mathfrak {v}+4 \mathfrak {q}\mathfrak {r}^{2} \mathfrak {s}^{2} \mathfrak {v}-2 \mathfrak {q}\mathfrak {s}^{2} \mathfrak {u}^{2} \mathfrak {v}-3 \mathfrak {r}^{2} \mathfrak {s}\mathfrak {u}^{2} \mathfrak {v}-\mathfrak {r}^{4} \mathfrak {s}\mathfrak {v}-2 \mathfrak {q}\mathfrak {r}\mathfrak {s}^{2} \mathfrak {u}\mathfrak {v}+3 \mathfrak {r}^{3} \mathfrak {s}\mathfrak {u}\mathfrak {v}\nonumber \\&\qquad +\mathfrak {q}^{3} \mathfrak {r}\mathfrak {s}\mathfrak {u}\mathfrak {v}+\mathfrak {r}^{5} \mathfrak {w}-5 \mathfrak {r}\mathfrak {s}^{3} \mathfrak {w}-\mathfrak {r}^{2} \mathfrak {u}^{3} \mathfrak {w}+2 \mathfrak {q}\mathfrak {s}\mathfrak {u}^{3} \mathfrak {w}+\mathfrak {r}\mathfrak {v}^{3} \mathfrak {w}+5 \mathfrak {q}^{2} \mathfrak {r}\mathfrak {s}^{2} \mathfrak {w}+3 \mathfrak {r}^{3} \mathfrak {u}^{2} \mathfrak {w}-6 \mathfrak {q}\mathfrak {r}\mathfrak {s}\mathfrak {u}^{2} \mathfrak {w}+2 \mathfrak {q}^{2} \mathfrak {r}\mathfrak {v}^{2} \mathfrak {w}-7 \mathfrak {r}\mathfrak {s}\mathfrak {v}^{2} \mathfrak {w}\nonumber \\&\qquad +4 \mathfrak {s}\mathfrak {u}\mathfrak {v}^{2} \mathfrak {w}-5 \mathfrak {q}\mathfrak {r}^{3} \mathfrak {s}\mathfrak {w}-3 \mathfrak {r}^{4} \mathfrak {u}\mathfrak {w}+4 \mathfrak {s}^{3} \mathfrak {u}\mathfrak {w}-\mathfrak {q}^{3} \mathfrak {r}^{2} \mathfrak {u}\mathfrak {w}-8 \mathfrak {q}^{2} \mathfrak {s}^{2} \mathfrak {u}\mathfrak {w}+2 \mathfrak {q}^{4} \mathfrak {s}\mathfrak {u}\mathfrak {w}+9 \mathfrak {q}\mathfrak {r}^{2} \mathfrak {s}\mathfrak {u}\mathfrak {w}+4 \mathfrak {q}\mathfrak {r}^{3} \mathfrak {v}\mathfrak {w}+11 \mathfrak {r}\mathfrak {s}^{2} \mathfrak {v}\mathfrak {w}\nonumber \\&\qquad +\mathfrak {q}\mathfrak {r}\mathfrak {u}^{2} \mathfrak {v}\mathfrak {w}+\mathfrak {q}^{4} \mathfrak {r}\mathfrak {v}\mathfrak {w}-3 \mathfrak {q}^{2} \mathfrak {r}\mathfrak {s}\mathfrak {v}\mathfrak {w}-5 \mathfrak {q}\mathfrak {r}^{2} \mathfrak {u}\mathfrak {v}\mathfrak {w}-8 \mathfrak {s}^{2} \mathfrak {u}\mathfrak {v}\mathfrak {w}+4 \mathfrak {q}^{2} \mathfrak {s}\mathfrak {u}\mathfrak {v}\mathfrak {w}. \end{aligned}$$
(A13e)
Similar to the previous step, it is now necessary to solve the equations \(d_{1}=d_{2}=0\) and the extra equation \(3 \mathfrak {q}\mathfrak {u}+ 4 \mathfrak {p}\mathfrak {v}+ 5 \mathfrak {w}=0\) extracted from Eq. (A13b), for the parameters \(\mathfrak {p}, \mathfrak {q}\) and \(\mathfrak {s}\). These equations result in the values
$$\begin{aligned} \mathfrak {p}= & {} {\frac{1}{54 \mathfrak {u}^{4}+600 \mathfrak {u}\mathfrak {v}\mathfrak {w}-320 \mathfrak {v}^{3}}}\left[ \mp \left( 27 \mathfrak {u}^{3} \mathfrak {v}+375 \mathfrak {u}\mathfrak {w}^{2}-400 \mathfrak {v}^{2} \mathfrak {w}\right) +\mathfrak {Q}\right] , \end{aligned}$$
(A14)
$$\begin{aligned} \mathfrak {q}= & {} {\frac{1}{27 \mathfrak {u}^{5} - 160 \mathfrak {u}\mathfrak {v}^{3} + 300 \mathfrak {u}^{2} \mathfrak {v}\mathfrak {w}}}\left[ 18 \mathfrak {u}^{3} \mathfrak {v}^{2} - 45 \mathfrak {u}^{4} \mathfrak {w}- 250 \mathfrak {u}\mathfrak {v}\mathfrak {w}\pm {\frac{2}{3}}\mathfrak {v}\mathfrak {Q}\right] , \end{aligned}$$
(A15)
$$\begin{aligned} \mathfrak {s}= & {} {\frac{1}{270 \mathfrak {u}^{4}+3000 \mathfrak {u}\mathfrak {v}\mathfrak {w}-1600 \mathfrak {v}^{3}}}\left[ 135 \mathfrak {u}^{4} \mathfrak {v}-1125 \mathfrak {u}^{2} \mathfrak {w}^{2}+3600 \mathfrak {u}\mathfrak {v}^{2} \mathfrak {w}-1280 \mathfrak {v}^{4}\mp 3\mathfrak {u}\mathfrak {Q}\right] , \end{aligned}$$
(A16)
where \(\mathfrak {Q} = 3|\mathfrak {u}|\sqrt{5 \left( -27 \mathfrak {u}^{4} \mathfrak {v}^{2}+2250 \mathfrak {u}^{2} \mathfrak {v}\mathfrak {w}^{2}+108 \mathfrak {u}^{5} \mathfrak {w}-1600 \mathfrak {u}\mathfrak {v}^{3} \mathfrak {w}+256 \mathfrak {v}^{5}+3125 \mathfrak {w}^{4}\right) }\). Note that, this process leaves \(\mathfrak {r}\) as a free parameter. This parameter can be however determined appropriately, by means of the equation \(d_{3} = 0\), which results in the cubic
$$\begin{aligned} e_{3} \mathfrak {r}^{3}+e_{2} \mathfrak {r}^{2}+e_{1} \mathfrak {r}+e_{0} = 0, \end{aligned}$$
(A17)
where
$$\begin{aligned}&e_{0} = 7 \mathfrak {p}^{2} \mathfrak {q}\mathfrak {u}\mathfrak {w}-4 \mathfrak {p}^{2} \mathfrak {q}\mathfrak {v}^{2}-9 \mathfrak {p}^{2} \mathfrak {s}\mathfrak {u}^{2}+\mathfrak {p}^{2} \mathfrak {u}^{2} \mathfrak {v}+\mathfrak {p}^{3} \mathfrak {u}^{3}-3 \mathfrak {p}^{3} \mathfrak {v}\mathfrak {w}-5 \mathfrak {p}^{2} \mathfrak {w}^{2}-\mathfrak {p}\mathfrak {q}^{2} \mathfrak {u}\mathfrak {v}-15 \mathfrak {p}\mathfrak {q}\mathfrak {s}\mathfrak {w}-3 \mathfrak {p}\mathfrak {q}\mathfrak {u}^{3}+2 \mathfrak {p}\mathfrak {q}\mathfrak {v}\mathfrak {w}\nonumber \\&\qquad +6 \mathfrak {s}^{2} (3 \mathfrak {p}\mathfrak {u}+4 \mathfrak {v})-15 \mathfrak {p}\mathfrak {s}\mathfrak {u}\mathfrak {v}-\mathfrak {p}\mathfrak {u}^{2} \mathfrak {w}+\mathfrak {p}\mathfrak {u}\mathfrak {v}^{2}-6 \mathfrak {q}^{2} \mathfrak {s}\mathfrak {v}-\mathfrak {q}^{3} \mathfrak {u}^{2}-8 \mathfrak {q}^{2} \mathfrak {u}\mathfrak {w}+4 \mathfrak {q}^{2} \mathfrak {v}^{2}+9 \mathfrak {q}\mathfrak {s}\mathfrak {u}^{2}-2 \mathfrak {q}\mathfrak {u}^{2} \mathfrak {v}-5 \mathfrak {q}\mathfrak {w}^{2}\nonumber \\&\qquad +12 \mathfrak {s}\mathfrak {u}\mathfrak {w}-18 \mathfrak {s}\mathfrak {v}^{2}-\mathfrak {u}^{4}-8 \mathfrak {u}\mathfrak {v}\mathfrak {w}+4 \mathfrak {v}^{3}-10 \mathfrak {s}^{3}, \end{aligned}$$
(A18a)
$$\begin{aligned}&e_{1} = -9 \mathfrak {q}\mathfrak {s}\mathfrak {u}+ 3 \mathfrak {p}\mathfrak {q}\mathfrak {u}^{2} + 3 \mathfrak {u}^{3} - 12 \mathfrak {p}\mathfrak {s}\mathfrak {v}+ 5 \mathfrak {p}^{2} \mathfrak {u}\mathfrak {v}- 2 \mathfrak {q}\mathfrak {u}\mathfrak {v}+ 8 \mathfrak {p}\mathfrak {v}^{2} + 5 \mathfrak {q}^{2} \mathfrak {w}- 15 \mathfrak {s}\mathfrak {w}- \mathfrak {p}\mathfrak {u}\mathfrak {w}+ 11 \mathfrak {v}\mathfrak {w}, \end{aligned}$$
(A18b)
$$\begin{aligned}&e_{2} = -3 \mathfrak {u}^{2} + 4 \mathfrak {q}\mathfrak {v}+ 5 \mathfrak {p}\mathfrak {w}, \end{aligned}$$
(A18c)
$$\begin{aligned}&e_{3} = \mathfrak {u}, \end{aligned}$$
(A18d)
whose solution, as it is well known, can be expressed in terms of radicals. Now, applying these solutions for \(\mathfrak {r}\), together with those expressed in Eqs. (A14)–(A16) for \(\mathfrak {p}\), \(\mathfrak {q}\) and \(\mathfrak {s}\), the values of \(d_{4,5}\) in Eqs. (A13d) and (A13e) are obtained. The expressions are, however, that huge that cannot be put in the paper. But we can be confident that the quintic (A12) has been reduced to the Bring–Jerrard form
$$\begin{aligned} z^{5} + d_{4} z+ d_{5} =0. \end{aligned}$$
(A19)
It is still possible to make more simplifications by defining
$$\begin{aligned} z\doteq {\frac{\mathfrak {t}}{\mathfrak {f}}}. \end{aligned}$$
(A20)
This way, the quintic (A19) can be recast as
$$\begin{aligned} \mathfrak {t}^{5}+d_{4} \mathfrak {f}^{4} \mathfrak {t}+d_{5}\mathfrak {f}^{5}=0. \end{aligned}$$
(A21)
Now letting
$$\begin{aligned} \mathfrak {f}=\left( \pm {\frac{1}{d_{4}}}\right) ^{\frac{1}{4}}, \end{aligned}$$
(A22)
we get to the more simplified Bring–Jerrard form of the quintic
$$\begin{aligned} \mathfrak {t}^{5}\pm \mathfrak {t}+K=0, \end{aligned}$$
(A23)
where we have defined \(K=d_{5} \mathfrak {f}^{5}\).
Appendix B: Derivation of the solutions to the x-parameter
Let us denote the solutions in the Eqs. (47)–(51) by \(\mathfrak {t}_{j}\) with \(j=\overline{1,5}\). Based on the definition in Eq. (A20), we have \(z_{j} = \mathfrak {f}^{-1} \mathfrak {t}_{j}\). Then from Eq. (A11), one needs to solve a quartic of the general form, in order to obtain an expression for \(y_{j}\) in terms of \(z_{j}\). This way, for each of the solutions for \(z_{j}\), we have four solutions for \(y_{j}\). To proceed with solving the quartic (A11), let us first apply the change of variable
$$\begin{aligned} y_{j}=W_{j}-{\frac{\mathfrak {p}}{4}}, \end{aligned}$$
(B1)
which depresses the equation to
$$\begin{aligned} W_{j}^{4}+\mathcal {A}W_{j}^{2}+\mathcal {B}W_{j}+\mathcal {C} = 0, \end{aligned}$$
(B2)
where
$$\begin{aligned}&\mathcal {A} = \mathfrak {q}-{\frac{3\mathfrak {p}^{2}}{8}}, \end{aligned}$$
(B3a)
$$\begin{aligned}&\mathcal {B} = \mathfrak {r}+{\frac{\mathfrak {p}^{3}}{8}}-{\frac{\mathfrak {p}\mathfrak {q}}{2}}, \end{aligned}$$
(B3b)
$$\begin{aligned}&\mathcal {C} = (\mathfrak {s}-z_{j})+{\frac{\mathfrak {p}^{2}\mathfrak {q}}{16}}-{\frac{3\mathfrak {p}^{4}}{256}}-{\frac{\mathfrak {p}\mathfrak {r}}{4}}. \end{aligned}$$
(B3c)
The method of solving the suppressed quartic (B2) has been given in the appendix C of Ref. [65]. Pursuing this method, we obtain the four solutions
$$\begin{aligned} W_{j1}= & {} \tilde{\mathcal {A}} + \sqrt{\tilde{\mathcal {A}^{2}}-\tilde{\mathcal {B}}}, \end{aligned}$$
(B4)
$$\begin{aligned} W_{j2}= & {} \tilde{\mathcal {A}} - \sqrt{\tilde{\mathcal {A}^{2}}-\tilde{\mathcal {B}}}, \end{aligned}$$
(B5)
$$\begin{aligned} W_{j3}= & {} - \tilde{\mathcal {A}} + \sqrt{\tilde{\mathcal {A}^{2}}-\tilde{\mathcal {C}}}, \end{aligned}$$
(B6)
$$\begin{aligned} W_{j4}= & {} - \tilde{\mathcal {A}} - \sqrt{\tilde{\mathcal {A}^{2}}-\tilde{\mathcal {C}}}, \end{aligned}$$
(B7)
in which
$$\begin{aligned}&\tilde{\mathcal {A}} = \sqrt{\tilde{\mathcal {U}}-{\frac{\mathcal {A}}{6}}}, \end{aligned}$$
(B8a)
$$\begin{aligned}&\tilde{\mathcal {B}} = 2\tilde{\mathcal {A}}^{2}+{\frac{\mathcal {A}}{2}}+{\frac{\mathcal {B}}{4\tilde{\mathcal {A}}}}, \end{aligned}$$
(B8b)
$$\begin{aligned}&\tilde{\mathcal {C}} = 2\tilde{\mathcal {A}}^{2}+{\frac{\mathcal {A}}{2}}-{\frac{\mathcal {B}}{4\tilde{\mathcal {A}}}}, \end{aligned}$$
(B8c)
where
$$\begin{aligned} \tilde{\mathcal {U}} = \sqrt{\frac{\tilde{\epsilon }_{2}}{3}}\cosh \left( {\frac{1}{3}}{\text {arccosh}}\left( 3\tilde{\epsilon }_{3}\sqrt{\frac{3}{\tilde{\epsilon }_{2}^{3}}}\right) \right) , \end{aligned}$$
(B9)
with
$$\begin{aligned}&\tilde{\epsilon }_{2}={\frac{\mathcal {A}^{2}}{12}}+\mathcal {C}, \end{aligned}$$
(B10a)
$$\begin{aligned}&\tilde{\epsilon }_{3} = {\frac{\mathcal {A}^{3}}{216}}-{\frac{\mathcal {A}\mathcal {C}}{6}}+{\frac{\mathcal {B}^{2}}{16}}. \end{aligned}$$
(B10b)
Finally, the solutions to the y-parameter are given as
$$\begin{aligned} \left( y_{j}\right) _{i}\equiv y_{ji} = W_{ji}-{\frac{\mathfrak {p}}{4}}, \end{aligned}$$
(B11)
where \(i=\overline{1,4}\). In this manner, the \(y_{ji}\) solutions form a \(5\times 4\) matrix (or in other words, four sets of solutions to the quintic (A7), in accordance with the solutions to the quintic (A12)). Now, in order to obtain the solutions for the x-parameter, as it is the original purpose of this discussion, we have to solve the quadratic equation (A2), for the known solutions \(y_{ji}\). This results in the two solutions
$$\begin{aligned} \left( x_{ji}\right) _{1}= & {} {\frac{-b_{1}+\sqrt{b_{1}^{2}-4\left( b_{2}-y_{ji}\right) }}{2}}, \end{aligned}$$
(B12)
$$\begin{aligned} \left( x_{ji}\right) _{2}= & {} {\frac{-b_{1}-\sqrt{b_{1}^{2}-4\left( b_{2}-y_{ji}\right) }}{2}}, \end{aligned}$$
(B13)
for \(b_{1,2}\) given in Eqs. (A5) and (A6). In this sense, for each of the \(y_{ji}\) solutions, there are two solutions for the x-parameter, which can be abbreviated as \(x_{jil}\) with \(l=1,2\). These solutions form a \(5\times 4\) matrix of \(2\times 1\) matrices, in the form
$$\begin{aligned} x_{jil}=\begin{pmatrix} z_{1}: \qquad y_{11}\rightarrow \begin{pmatrix} x_{111}\\ x_{112} \end{pmatrix} &{} y_{12}\rightarrow \begin{pmatrix} x_{121}\\ x_{122} \end{pmatrix} &{} y_{13}\rightarrow \begin{pmatrix} x_{131}\\ x_{132} \end{pmatrix} &{} y_{14}\rightarrow \begin{pmatrix} x_{141}\\ x_{142} \end{pmatrix}\\ z_{2}: \qquad y_{21}\rightarrow \begin{pmatrix} x_{211}\\ x_{212} \end{pmatrix} &{} y_{22}\rightarrow \begin{pmatrix} x_{221}\\ x_{222} \end{pmatrix} &{} y_{23}\rightarrow \begin{pmatrix} x_{231}\\ x_{232} \end{pmatrix} &{} y_{24}\rightarrow \begin{pmatrix} x_{241}\\ x_{242} \end{pmatrix}\\ z_{3}:\qquad y_{31}\rightarrow \begin{pmatrix} x_{311}\\ x_{312} \end{pmatrix} &{} y_{32}\rightarrow \begin{pmatrix} x_{321}\\ x_{322} \end{pmatrix} &{} y_{33}\rightarrow \begin{pmatrix} x_{331}\\ x_{332} \end{pmatrix} &{} y_{34}\rightarrow \begin{pmatrix} x_{341}\\ x_{342} \end{pmatrix}\\ z_{4}:\qquad y_{41}\rightarrow \begin{pmatrix} x_{411}\\ x_{412} \end{pmatrix} &{} y_{42}\rightarrow \begin{pmatrix} x_{421}\\ x_{422} \end{pmatrix} &{} y_{43}\rightarrow \begin{pmatrix} x_{431}\\ x_{432} \end{pmatrix} &{} y_{44}\rightarrow \begin{pmatrix} x_{441}\\ x_{442} \end{pmatrix}\\ z_5:\qquad y_{51}\rightarrow \begin{pmatrix} x_{511}\\ x_{512} \end{pmatrix} &{} y_{52}\rightarrow \begin{pmatrix} x_{521}\\ x_{522} \end{pmatrix} &{} y_{53}\rightarrow \begin{pmatrix} x_{531}\\ x_{532} \end{pmatrix} &{} y_{54}\rightarrow \begin{pmatrix} x_{541}\\ x_{542} \end{pmatrix} \end{pmatrix}, \end{aligned}$$
(B14)
meaning that for the solutions of the \(\mathfrak {t}\)-parameter derived from the quintic (A23), there are eight sets of solutions for the x-parameter in the context of the original quintic (A1).
