\chapter{Table of $h_0(x)$ and $h_1(x)$}
Let $q=2^{10},\F_{q^2}^*=\langle \lambda \rangle$, and consider extensions of $\F_{q^{2k}}$ such that the extension degree is prime and
$512<k<1024$. We want to compute $h_0(x)$ and $h_1(x)$ over $\F_{q^2}[x]$ such that they satisfy the following properties:
\begin{itemize}
  \item $ \deg(h_0) \leq 2, \deg(h_1)\leq 1 $;
  \item $ k_i > 1 $ for all $ 1\leq i\leq l $; In other words, 
it is free of linear factors; 
  \item $ \gcd(k, k_i) =1 $  for all $ 1\leq i\leq l $. 
\end{itemize}

\newpage
\begin{table}[ht]
\centering
\begin{tabular} {|c|c|c|}
\hline \hline
Extension Degree&     h0            &          h1    \\ \hline
 521			&	$x^2 + \lambda^{333}$  	&  $x + \lambda$ \\ \hline
 523			&   $x^2 + \lambda^{768}$		& $x + \lambda$ \\ \hline
 541			&   $x^2 + \lambda^{2838}$	& $x + \lambda$ \\ \hline
 547			&   $x^2 + \lambda^{1616}$	& $x + \lambda$ \\ \hline
 557			&   $x^2 + \lambda^{2978}$	& $x + \lambda$ \\ \hline
 563			&   $x^2 + \lambda^{399}$		& $x + \lambda$ \\ \hline
 569			&	$x^2 + \lambda^{1266}$	& $x + \lambda$ \\ \hline
 571			&	$x^2 + \lambda^{527}$		& $x + \lambda$ \\ \hline
 577			&	$x^2 + \lambda^{437}$		& $x + \lambda$ \\ \hline
 587			&	$x^2 + \lambda^{284}$		& $x + \lambda$ \\ \hline
 593			&	$x^2 + \lambda^{843}$		& $x + \lambda$ \\ \hline
 599			&	$x^2 + \lambda^{1244}$	& $x + \lambda$ \\ \hline
 601			&	$x^2 + \lambda^{2216}$	& $x + \lambda$ \\ \hline
 607			&	$\lambda^7*x^2 + \lambda^{17}*x + \lambda^{406}$ & $x$ \\ \hline
 613			&  	$x^2 + \lambda^{296}$		& $x + \lambda$ \\ \hline
 617			&	$x^2 + \lambda^{1714}$	& $x + \lambda$ \\ \hline
 619			&	$x^2 + \lambda^{5864}$	& $x + \lambda$ \\ \hline
 631			&	$x^2 + \lambda^{682}$		& $x + \lambda$ \\ \hline
 641			&	$x^2 + \lambda^{4014}$	& $x + \lambda$ \\ \hline
 643			&	$x^2 + \lambda^{2592}$	& $x + \lambda$ \\ \hline
 647			&	$x^2 + \lambda^{600}$		& $x + \lambda$ \\ \hline
 653			&	$x^2 + \lambda^{397}$		& $x + \lambda$ \\ \hline
 659			&	$x^2 + \lambda^{717}$		& $x + \lambda$ \\ \hline
 661			&	$x^2 + \lambda^{5081}$	& $x + \lambda$ \\ \hline
 673			&	$x^2 + \lambda^{280}$		& $x + \lambda$ \\ \hline
 677			&	$x^2 + \lambda^{1797}$	& $x + \lambda$ \\ \hline
 683			&	$x^2 + \lambda^{302}$		& $x + \lambda$ \\ \hline
 691			&	$x^2 + \lambda^{3802}$	& $x + \lambda$ \\ \hline
 701			&	$x^2 + \lambda^{1934}$	& $x + \lambda$ \\ \hline
 709			&	$x^2 + \lambda^{3655}$	& $x + \lambda$ \\ \hline
 719			&	$x^2 + \lambda^{2543}$	& $x + \lambda$ \\ \hline
 727			&	$x^2 + \lambda^{3964}$	& $x + \lambda$ \\ \hline
 733			&	$x^2 + \lambda^{4427}$	& $x + \lambda$ \\ \hline
 739			&	$x^2 + \lambda^{4661}$	& $x + \lambda$ \\ \hline
 743			&	$x^2 + \lambda^{500}$		& $x + \lambda$ \\ \hline
 751			&	$x^2 + \lambda^{1405}$	& $x + \lambda$ \\ \hline
 757			&	$x^2 + \lambda^{1317}$	& $x + \lambda + 1$ \\ \hline
  \hline
  \end{tabular}
  \caption{Representation of $\F_{q^{2k}}$ for $q=2^{10}$ and prime extension degree}
  \end{table}
 
\newpage  
\begin{table}[ht]
\centering
\begin{tabular} {|c|c|c|}
\hline \hline
Extension Degree&     h0            &          h1    \\ \hline 
 761			&	$x^2 + \lambda^{3000}$	& $x + \lambda$ \\ \hline
 769			&	$x^2 + \lambda^{1645}$	& $x + \lambda$ \\ \hline
 773			&	$x^2 + \lambda^{1882}$	& $x + \lambda$ \\ \hline
 787			&	$x^2 + \lambda^{3996}$	& $x + \lambda$ \\ \hline
 797			&	$x^2 + \lambda^{2572}$	& $x + \lambda$\\ \hline
 809			&	$x^2 + \lambda^{551}$		&$x + \lambda$ \\ \hline
 811			&	$x^2 + \lambda^{101}$		& $x + \lambda$ \\ \hline
 821			&	$x^2 + \lambda^{1941}$	& $x + \lambda$ \\ \hline
 823			&	$x^2 + \lambda^{1220}$	& $x + \lambda + 1$ \\ \hline
 827			&	$x^2 + \lambda^{1607}$	& $x + \lambda$ \\ \hline
 829			&	$x^2 + \lambda^{2918}$	& $x + \lambda$ \\ \hline
 839			&	$x^2 + \lambda^{454}$		& $x + \lambda$ \\ \hline
 853			&	$x^2 + \lambda^{1386}$	& $x + \lambda$ \\ \hline
 857			&	$x^2 + \lambda^{1493}$	& $x + \lambda$ \\ \hline
 859			&	$x^2 + \lambda^{404}$		& $x + \lambda$ \\ \hline
 863			&	$x^2 + \lambda^{1063}$	& $x + \lambda$ \\ \hline
 877			&	$x^2 + \lambda^{1898}$	&$x + \lambda$ \\ \hline
 881			&	$x^2 + \lambda^{383}$		& $x + \lambda$ \\ \hline
 883			&	$x^2 + \lambda^{1824}$	& $x + \lambda$ \\ \hline
 887			&	$x^2 + \lambda^{4621}$	& $x + \lambda$ \\ \hline
 907			&	$x^2 + \lambda^{40}$		& $x + \lambda$ \\ \hline
 911			&	$x^2 + \lambda^{1784}$	& $x + \lambda$ \\ \hline
 919			&	$x^2 + \lambda^{4419}$	& $x + \lambda$ \\ \hline
 929			&	$x^2 + \lambda^{1119}$	& $x + \lambda$ \\ \hline
 937			&	$x^2 + \lambda^{2433}$	& $x + \lambda$ \\ \hline
 941			&	$x^2 + \lambda^{132}$		& $x + \lambda$ \\ \hline
 947			&	$x^2 + \lambda^{1078}$	& $x + \lambda$ \\ \hline
 953			&	$x^2 + \lambda^{1032}$	& $x + \lambda$ \\ \hline
 967			&	$x^2 + \lambda^{1509}$	& $x + \lambda$\\ \hline
 971			&	$x^2 + \lambda^{1712}$	& $x + \lambda$ \\ \hline
 977			&	$x^2 + \lambda^{2460}$	& $x + \lambda$ \\ \hline
 983			&	$x^2 + \lambda^{2864}$	& $x + \lambda$ \\ \hline
 991			&	$x^2 + \lambda^{1310}$	& $x + \lambda + 1$ \\ \hline
 997			&	$x^2 + \lambda^{1421}$	& $x + \lambda$\\ \hline
 1009			&	$x^2 + \lambda^{3152}$	& $x + \lambda$ \\ \hline
 1013			&	$x^2 + \lambda^{738}$		&$x + \lambda$ \\ \hline
 1019			&	$x^2 + \lambda^{4900}$	& $x + \lambda$\\ \hline
 1021			&	$x^2 + x + \lambda^{306}$	& $x + \lambda$ \\ \hline
 \hline
\end{tabular}
\caption{Representation of $\F_{q^{2k}}$ for $q=2^{10}$ and prime extension degree (continued)}
\end{table}
