\documentclass{amsart}
\def\KK{\mathbb K}

\begin{document}

Setting $X=x+\frac{1}{x}$, we have
\[
X^3 + C_n X^2  - (C_n-6) X -(C_n^2+2C_n-16)=0,
\]
and with $X=Y-C_n/3$, we obtain
\[
Y^3+(6-C_n-C_n^2/3)Y+(16-4C_n-2C_n^2/3+2C_n^3/27)=0.
\]
Put $Y=u+v$, $p=6-C_n-C_n^2/3$ and
$q=16-4C_n-2C_n^2/3+2C_n^3/27$. Thus $u^3$ and $v^3$ are solutions of
$Z^2+qZ-p^3/27=0,$ and so
\[
u=\sqrt[3]{-27q/2\!+\!3\sqrt{3\Delta}/2},\;a_1=\frac{u\!+\!\overline{u}\!-\!C_n}{3},\;
a_2=\frac{ju\!+\!\overline{ju}\!-\!C_n}{3},\;a_3=\frac{j^2u\!+\!\overline{j^2u}\!-\!C_n}{3},
\]
where $\overline{u}$ is the complex conjugate of $u$. Therefore
\[
\nu_1=\frac{a_2-\sqrt{a_2^2-4}}{2},\;\;
\nu_2=\frac{a_1+\sqrt{a_1^2-4}}{2},\;\; \nu_3 =\frac{a_3+\sqrt{a_3^2-4}}{2},
\]
\[
\nu_4=\frac{a_2+\sqrt{a_2^2-4}}{2},\;\;
\nu_5=\frac{a_1-\sqrt{a_1^2-4}}{2},\;\; \nu_6 =\frac{a_3-\sqrt{a_3^2-4}}{2},
\]
whereby $\varepsilon=|\nu_1|,\; \varepsilon'=|\nu_2|.$



We determine the constants $X_0,\, X_1,\, X_2,\, X_3$,
$c_1,\dots , c_{12}$, $\delta_i$ and $\lambda_i$:
\[
X_0=1, \,
c_1=\frac{16\theta(\theta-1)(\theta+1)(\theta+2)(2\theta+1)^5}{9(\theta^2+\theta+1)^5},\,
c_2=\frac{\theta^2+\theta+1}{(2\theta+1)(\theta+1)},
\]
\[
X_1=\max(X_0, (7c_1/c_2)^{1/6}),\;
c_3=\frac{7}{6(\theta^2+\theta+1)}c_1, \; c_4=1.39c_3,
X_2=\max(X_1, (2c_3)^{1/6}),
\]

$\begin{array}{lcl}
\vspace{5mm}

c_5 =  \left\{ \begin{array}{ll}
\vspace{5mm}

\left|\frac{\log(\varepsilon_3^2 \varepsilon_3')}{U}\right| & if\; n=0,\\

\left|\frac{\log(\varepsilon^2/ \varepsilon')}{V}\right| & \mbox{otherwise},
           \end{array}
    \right.
\end{array}$

$\begin{array}{lcl}
\vspace{5mm}

c_6 \!=\!
\!\left\{ \!\begin{array}{l}
\vspace{5mm}

-\frac{\log(\varepsilon_3^2\varepsilon_3')}{6U}\!+\!\left|\frac{-\log(\varepsilon_3)\log(T_2T_5)-\log(\varepsilon_3\varepsilon_3')\log(T_3T_6)}{2U}\right| \;\;\mbox{if } n=0,\\

\frac{\log(\varepsilon^2/\varepsilon')}{6V}\!+\!\left|\frac{\log((\varepsilon')^2/\varepsilon)\log(T_2)\!+\!3\log(\varepsilon'/\varepsilon)\log(T_3)\!+\!2\log(\varepsilon'/\varepsilon^2)\log(T_4)\!-\!3\log(\varepsilon)\log(T_5)\!-\!\log(\varepsilon\varepsilon')\log(T_6)}{6V}\right|,
           \end{array}
    \right.
\end{array}$

where
\[
T_j=\theta-\theta_j,\;\; j\neq 1,\;\;
U=\log^2(\varepsilon_3)+\log^2(\varepsilon_3')+\log(\varepsilon_3)\log(\varepsilon_3'),
\]
\[
V=\log^2(\varepsilon)+\log^2(\varepsilon')-\log(\varepsilon)\log(\varepsilon'),
\]
\[
c_7=c_5,\;\;\; c_8=c_6,\;\;\; c_9=c_4 \exp(6c_6/c_5),\;\;\;
c_{10}=6/c_5
\]
because $\KK_t$ is real;
\[
c_{11}=7\times 5\times 3^{14}\times
2^{59} \log(72) \log\left(\varepsilon_2\right)
\log^2(\varepsilon_3^{-2}(\varepsilon_3')^{-1})
\log^2(\varepsilon) \log[9(\theta+2)],
\]
\[
B_0=\max\left(e,2c_{10}^{-1}c_{11}\log\left(c_{10}^{-1}c_{11}c_9^{1/c_{11}}\right)\right),
\]
\[
i_1=\left\{ \begin{array}{ll} 2 & if\; n=0,\\

4 & \mbox{otherwise},
           \end{array}
    \right.
\;\;\; i_2=1,
\]
\[
\delta=\left\{ \begin{array}{ll} \vspace{3mm}

\delta_1/\delta_2 & if\; n=0,\\

\delta_1/\delta_4 & \mbox{otherwise},
           \end{array}
    \right.
\;\;\;
\lambda=\left\{ \begin{array}{ll}
\vspace{3mm}

(\delta_1\lambda_2-\delta_2\lambda_1)/\delta_2 & if\; n=0,\\

(\delta_1\lambda_4-\delta_4\lambda_1)/\delta_4 & \mbox{otherwise},
           \end{array}
    \right.
\]
\[
 c_{14}=c_{12} \exp(6c_6/c_5), c_{15}=6/c_5,
\]
\[
\left\{ \begin{array}{lll} \vspace{3mm}

\delta_1=-\frac{1}{\log(\varepsilon_2)},  & \delta_2=-\frac{\log(\varepsilon_3^2 \varepsilon_3')}{U}, & \delta_3=-\frac{\log(\varepsilon_3 (\varepsilon_3')^2)}{U},\\

 \delta_4=-\frac{\log(\varepsilon^2/\varepsilon')}{V}, & \delta_5=\frac{\log(\varepsilon/(\varepsilon')^2)}{V}, &
           \end{array}
    \right.
\]
and
\[
\left\{ \!\begin{array}{l} \vspace{3mm}

\lambda_1=-\frac{\log(T_2T_4T_6)}{3\log(\varepsilon_2)},\\
 \vspace{3mm}

\lambda_2=\frac{-\log(\varepsilon_3)\log(T_2T_5)-\log(\varepsilon_3\varepsilon_3')\log(T_3T_6)}{2U},\\
\vspace{3mm}

\lambda_3=\frac{-\log(\varepsilon_3\varepsilon_3')\log(T_2T_5)-\log(\varepsilon_3')\log(T_3T_6)}{2U},\\
\vspace{3mm}

\lambda_4\!=\!\frac{\log((\varepsilon')^2/\varepsilon)\log(T_2)+3\log(\varepsilon'/\varepsilon)\log(T_3)+2\log(\varepsilon'/\varepsilon^2)\log(T_4)-3\log(\varepsilon)\log(T_5)-\log(\varepsilon\varepsilon')\log(T_6)}{6V},\\

\lambda_5\!=\!\frac{-\log(\varepsilon\varepsilon')\log(T_2)-3\log(\varepsilon')\log(T_3)-2\log((\varepsilon')^2/\varepsilon)\log(T_4)-3\log(\varepsilon'/\varepsilon)\log(T_5)-\log(\varepsilon'/\varepsilon^2)\log(T_6)}{6V}.
           \end{array}
    \right.
\]
Using the following system
\[
\left\{ \begin{array}{l}

\gamma_1=x-\theta y=\varepsilon_2^{b_1} \varepsilon_3^{b_2} (\varepsilon_3')^{b_3} \varepsilon^{b_4} (\varepsilon')^{b_5},\\

\gamma_2=x-\theta_2 y=\varepsilon_2^{-b_1} (\varepsilon_3')^{b_2} \left(\varepsilon_3\varepsilon_3'\right)^{-b_3} (\varepsilon')^{b_4} \left(-\varepsilon^{-1}\varepsilon'\right)^{b_5},
\end{array}
    \right.
\]
we obtain
\[
\left\{ \begin{array}{l}
\vspace{5mm}

x= \frac{-(\theta-1)\gamma_1 + \theta(\theta+2)\gamma_2}{\theta^2+\theta+1},\\

y= \frac{(\theta+2)(-\gamma_1+\gamma_2)}{\theta^2+\theta+1}.
\end{array}
    \right.
\]


The computations are done with a SUN PARC
ULTRA1, and for each value of $n$, the time of computation is roughly 9
seconds.

\end{document}