\section{The data of the sieve}\label{app:data}

This appendix prints the data of the sieve: the basis of $\Sel$, the elements $u_i$ and the norm matrix, the primes of $\Lt$ above the auxiliary primes, the symbol matrices $\Mq$, the sets $\Iq$, the affine space $\mathcal A$ of \cref{thm:sieve} and the last stages of the sieve.  The same data, in machine-readable form, are part of the deposited programs and data~\cite{Companion257}, and the tables below are generated from them.  Throughout, $a$ is a root of $h$ and $b$ a root of $\psi$, as in \cref{sec:fields}.

\subsection{The basis of \texorpdfstring{$\Sel$}{L24(S,5)}}\label{app:B}

\Cref{tab:B} lists $B_1,\dots,B_{24}$.  Any other elements with the same classes modulo $\Lt^{\times5}$ would serve equally well.  All $B_j$ are algebraic integers, since their valuations are nonnegative at every prime.  The denominators $D_j$ appear because the ring of integers of $\Lt$ is not contained in $\Z[a,b]$ ($41$ divides the index of $\Z[a]$ in the ring of integers of $\Lo$, and $\Lo$ and $\Q(b)$ are both ramified at $2$, $5$ and $7$).

\input{data/tab-B}

\subsection{The norm data}\label{app:u}

\Cref{tab:u} lists $u_1,\dots,u_{10}\in\Lo$, and~\eqref{eq:NM} gives the matrix $\NM$ and the vector $\TT$ of \cref{lem:normdata}: column $j$ of $\NM$ is the exponent vector of $\Norm{\Lt/\Lo}{B_j}$ on $u_1,\dots,u_{10}$, up to sign, and $\TT$ is that of $2$.
\input{data/tab-u}
\input{data/NM}

\subsection{The primes above the auxiliary primes}\label{app:primesq}

\Cref{tab:primesq} describes the primes $\qQ_1,\dots,\qQ_m$ of $\Lt$ above each $q\in\cQ$ by their residue maps.  Since $q\nmid2\cdot5\cdot7\cdot41$, the prime $\qq=\qQ\cap\Lo$ is $(q,h_k(a))$ for an irreducible factor $h_k$ of $h\bmod q$ (the Dedekind--Kummer theorem), $b$ is $\qQ$-integral, and the primes of $\Lt$ above $\qq$ correspond to the irreducible factors of $\psi$ over $k(\qq)$.  To compute $\sym_{\qQ_k}(B_j)$ from the table, reduce the coefficients of $B_j$ modulo $q$ (they are $q$-integral), map $a$ to $x$ in $\Fq q[x]/(h_k)$ and $b$ to $r(x)$, or, in the rows marked ``cubic'' or ``quadratic'', to a root of that factor of $\psi$ over $\Fq q[x]/(h_k)$, and apply~\eqref{eq:symbol}.  The symbol does not depend on which root is chosen.

\input{data/tab-primes}

\subsection{The symbol matrices}\label{app:Mq}

\input{data/tab-Mq}

\subsection{The sets \texorpdfstring{$\Iq$}{I(q)}}\label{app:Iq}

\input{data/tab-Iq}

\subsection{The affine space and the last stages of the sieve}\label{app:affine}

The affine space $\mathcal A$ of \cref{thm:sieve}, cut out by (N) and (V), is $\{c_0+\sum_{k=1}^{10}\lambda_kd_k:\lambda\in\F_5^{10}\}$ with
\input{data/affine}%
written in blocks of six coordinates.  \Cref{tab:survivors} lists the vectors that survive the conditions at $181$, $311$ and $131$.

\input{data/tab-survivors}