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\section{The computations}\label{sec:computations}
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Our computations were performed in Magma V2.30-1 \cite{magma}, apart from some auxiliary ones in Python and in
SageMath 10.7 (such as the lists of groups, the values of Abramovich's bound, the modular units of
\Cref{prop:families}, which were found with the package \texttt{mdsage} \cite{mdsage}, and the replay of data exported
from Magma). They were run on the \textit{Mordell} server at the Department of Mathematics, University of Zagreb,
with an AMD EPYC 9175F CPU and $384$ GB of RAM, between $29$ and $31$ October 2026. We used the package
\texttt{mdmagma} \cite{mdmagma} for the moduli-theoretic operations, Sutherland's optimized models of $X_1(n)$
\cite{sutherland,sutherlandtables}, the Derickx-Sutherland models of $X_1(m,n)$ \cite{DSmodels}, the low degree points
of van Hoeij \cite{vanHoeij}, and the LMFDB \cite{lmfdb} for the analytic ranks of newforms. The jobs recorded in the
repository took about $66$ hours of CPU time in total. This includes exploratory and diagnostic runs which the proofs
do not use, and does not include the computations from \cite{N69} that we reuse. Those are copied into the repository
together with the scripts that produced them.

Several results were also checked by computations that their proofs do not need. They are listed, result by
result and with their logs, in the section ``Further checks'' of the \texttt{README} file of the repository. Most of
them repeat a computation by an independent program, usually in SageMath or Python with integer or finite field
linear algebra, starting from the data exported by the Magma computation. For example, the $1086016$ restriction
ranks for $\tg{3}{33}$, the $262$
parity decisions for $\tg{4}{16}$, the classes and matches in the proofs for $\tg{3}{15}$, $\tg{4}{16}$ and
$\tg{2}{2n}$, the determinants and norms in \Cref{lem:glob-odd}, and the linear independence of the vectors
$v,vT_2,\dots$ in the winding computations of \Cref{sec:global} were all recomputed in this way. Other results were
checked by a second method or at other primes: for example the isogenies used in the Hecke operators were
recomputed from the division polynomials with V\'elu's formulas, the Riemann-Roch computations at $13$ for
$\tg{6}{12}$ were repeated at $19$ and $31$, and the divisor classes were recomputed from ideal identities in the
function fields. In all cases the results agree.

A few computations were made for the final version of the paper, when some of the original arguments were replaced
by shorter ones. They are in the folder \githubdir{additions} of the repository, and they are linked in the proofs
of \Cref{prop:4x20,prop:56,lem:sw-60-quintic,prop:74} and in \Cref{sec:sporadic}. Some arguments of the original
computational record are therefore no longer needed in the paper: for example the formal immersion computations for
the entirely cuspidal reductions on $X_1(49)$, and the computation of the gonality of $X_1(4,20)$ over $\F_9$, which
needed $11156288$ rank computations.


We also checked some of the computations a second time, with code written independently
of the first implementation. These are the list of \Cref{lem:minimal}, which we \gitlink{independent_checks/minimal_groups.py}{recomputed}
from scratch, and the explicit examples of \Cref{prop:explicit_cyclic,prop:2x28-2x30}, for which we
\gitlink{independent_checks/occ_torsion_check.m}{computed} the torsion groups with Magma's function
\texttt{TorsionSubgroup}, which does not use the reductions of \Cref{table:explicit}, both from the models used in the
proofs and, for the points of \Cref{prop:explicit_cyclic}, \gitlink{independent_checks/vanhoeij_check.m}{directly} from
van Hoeij's coordinates. We also checked the field automorphism and the $j$-invariants used in \Cref{sec:sporadic}
(\githubbare{independent_checks/aut_318_check.m}, \githubbare{independent_checks/sporadic_check.m}). In all cases the
results agree. Finally, we checked that every file of the repository that is
used as evidence in the original record has the hash recorded there
(\githubbare{independent_checks/check_ledger_hashes.py}). The files that are only in the release archive of the
repository, or that are copies of other papers, are listed there.