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\newcommand{\Q}{\mathbb{Q}}
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\newcommand{\PP}{\mathbb{P}}
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\DeclareMathOperator{\gon}{gon}
\DeclareMathOperator{\Pic}{Pic}
\DeclareMathOperator{\Jac}{Jac}
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\newtheorem*{theoremA}{Theorem A}
\newtheorem*{theoremB}{Theorem B}
\newtheorem*{theoremC}{Theorem C}
\newtheorem*{theoremD}{Theorem D}
\newtheorem*{corollaryE}{Corollary E}
\newtheorem{thm}{Theorem}[section]
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\begin{document}

\title{Gonality and quintic points on intermediate modular curves}
\author{Marin Varivoda}
\address{Department of Mathematics, Faculty of Science, University of Zagreb, Bijeni\v{c}ka cesta 30, 10000 Zagreb, Croatia}
\email{marin.varivoda@math.hr}
\date{\today}
\subjclass[2020]{11G18, 14H51, 11G05, 14G05}
\keywords{modular curves, gonality, points of low degree, density degree}

\begin{abstract}
For $\{\pm1\}\subseteq\Delta\subseteq(\Z/N\Z)^\times$ let $X_\Delta(N)$ be the intermediate modular curve
between $X_1(N)$ and $X_0(N)$. We determine the $\Q$-gonality of every strict intermediate curve $X_\Delta(N)$
with $N\le 50$, and of $X_1(N)$ for $41\le N\le 50$, where it equals the upper bound of Derickx and van Hoeij
at every level. We also settle the thirty levels left open by Derickx, Hwang, Jeon and Orli\'c, completing the
list of curves $X_0(N)$ with infinitely many points of degree $5$, and we show that exactly $43$ strict
intermediate curves have infinitely many points of degree $5$. As a consequence, the strict intermediate curves
of density degree $5$ are exactly the four $\Q$-pentagonal curves found by Orli\'c. The proofs use published
classifications, the Castelnuovo--Severi inequality, and three computational methods; all computations can be
rerun from the accompanying code.
\end{abstract}

\maketitle

\section{Introduction}

Let $N$ be a positive integer and $\Delta$ a subgroup with $\{\pm1\}\subseteq\Delta\subseteq(\Z/N\Z)^\times$.
The intermediate modular curve $X_\Delta(N)$ is the modular curve attached to
\[
 \Gamma_\Delta(N)=\left\{\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\SL_2(\Z): c\equiv 0 \bmod N,\ a \bmod N\in\Delta\right\}.
\]
It lies between $X_1(N)=X_{\{\pm1\}}(N)$ and $X_0(N)=X_{(\Z/N\Z)^\times}(N)$, and it is defined over $\Q$.
We call $X_\Delta(N)$ \emph{strict} if both inclusions $\{\pm1\}\subsetneq\Delta\subsetneq(\Z/N\Z)^\times$
are strict. We follow the notation of Jeon, Kim and Schweizer \cite{JeonKimSchweizer2020}, Orli\'c
\cite{Orlic2025} and Derickx and Orli\'c \cite{DerickxOrlic2026}, and we name curves by their LMFDB labels
\cite{LMFDB,RSZB2022}.

For a curve $C$ over a field $k$, the \emph{$k$-gonality} $\gon_k C$ is the least degree of a nonconstant
morphism $C\to\PP^1$ defined over $k$. A \emph{point of degree $d$} on $C/\Q$ is a closed point whose residue
field has degree $d$ over $\Q$. The \emph{density degree} of $C$ is the least $d$ such that $C$ has
infinitely many points of degree $d$. By Hilbert irreducibility, $C$ has infinitely many points of degree
$\gon_\Q C$, so the density degree is at most the $\Q$-gonality.

\subsection{Previous work}
Points of low degree on modular curves have been studied intensively, with the degree going up one at a
time. For $X_0(N)$, the curves with infinitely many points of degree $2$ follow from Ogg's classification of
hyperelliptic curves \cite{Ogg1974} and Bars' classification of bielliptic curves \cite{Bars1999}; degree
$3$ is due to Jeon \cite{Jeon2021}; degree $4$ to Hwang and Jeon \cite{HwangJeon2024} and independently to
Derickx and Orli\'c \cite{DerickxOrlicX0}; and degree $5$ to Derickx, Hwang, Jeon and Orli\'c \cite{DHJO},
except for thirty levels that their methods did not reach. For intermediate curves, degree $2$ is due to
Jeon, Kim and Schweizer \cite{JeonKimSchweizer2020}, degree $3$ to Dalal \cite{Dalal2023}, and degree
$4$ to Derickx and Orli\'c \cite{DerickxOrlic2026}. In \cite{DerickxOrlic2026} the authors explain the obstacles to extending
their methods to degree $5$; Theorem~B below settles that case.

For gonality, the hyperelliptic and trigonal intermediate curves were found by Ishii and Momose
\cite{IshiiMomose1991} and Jeon and Kim \cite{JeonKim2007}. Orli\'c \cite{Orlic2025} determined all
intermediate curves of $\Q$-gonality at most $5$, and the $\Q$-gonality of every strict curve with $N\le40$.
The $\Q$-gonality of $X_0(N)$ is known for $N<145$ by Najman and Orli\'c \cite{NajmanOrlic2024}, and that
of $X_1(N)$ for $N\le40$ by Derickx and van Hoeij \cite{DerickxVanHoeij2014}, who also gave upper bounds
for $N\le 250$.

\subsection{Results}
\begin{theoremC}\label{thm:C}
Let $X_\Delta(N)$ be a strict intermediate curve with $N\le 50$. Its $\Q$-gonality is given in
\cref{tab:C}. For $N\le 40$ these values are due to Orli\'c \cite{Orlic2025}, as are those of $6$ of the
$24$ curves with $41\le N\le 50$, which have gonality $4$ or $5$; two more have genus $3$ and $4$ and are
trigonal. The values for the other $16$ curves are new.
\end{theoremC}

\begin{theoremD}\label{thm:D}
For $41\le N\le 50$, the $\Q$-gonality of $X_1(N)$ equals the upper bound of Derickx and van Hoeij
\cite{DerickxVanHoeij2014}:
\begin{center}
\begin{tabular}{l|cccccccccc}
$N$ & 41 & 42 & 43 & 44 & 45 & 46 & 47 & 48 & 49 & 50\\\hline
$\gon_\Q X_1(N)$ & 22 & 12 & 24 & 15 & 18 & 19 & 29 & 16 & 21 & 15
\end{tabular}
\end{center}
\end{theoremD}

\begin{theoremA}\label{thm:A}
For each of the thirty levels
\begin{align*}
 N\in\{&74, 77, 79, 82, 83, 85, 86, 88, 89, 92, 96, 98, 99, 100, 101,\\
 &103, 107, 111, 118, 123, 128, 131, 141, 142, 143, 145, 155, 159, 167, 191\}
\end{align*}
left open in \cite[Remark~1.11 and \S7]{DHJO}, the curve $X_0(N)$ has only finitely many points of degree $5$, and no
morphism of degree $5$ to $\PP^1$ over $\Q$. Together with \cite[Theorem~1.10]{DHJO}, this completes the
list of curves $X_0(N)$ with infinitely many points of degree $5$.
\end{theoremA}

\begin{theoremB}\label{thm:B}
Exactly $43$ strict intermediate curves $X_\Delta(N)$ have infinitely many points of degree $5$. They are
listed in \cref{tab:positive}, and each of them has a morphism of degree $5$ to $\PP^1$ over $\Q$.
\end{theoremB}

\begin{corollaryE}\label{cor:E}
The strict intermediate curves of density degree $5$ are exactly the four $\Q$-pentagonal curves of
Orli\'c \cite[Theorems~1.4--1.5]{Orlic2025}: $X_\Delta(31)$ with $\Delta=\langle-1,2\rangle$ and
$\Delta=\langle-1,5\rangle$, $X_\Delta(44)$ with $\Delta=\langle-1,5\rangle$, and $X_\Delta(125)$ with
$\Delta=\langle-1,4\rangle$.
\end{corollaryE}

This parallels the case of $X_0(N)$, where $X_0(109)$ is the only curve of density degree $5$ \cite{DHJO}.

\subsection{Methods}
We have tried to keep the new mathematical input to a minimum, so that the paper can be checked by a human
reader. Wherever a published result or an inequality settles a case, we use it: Castelnuovo--Severi, the
classifications of curves of small gonality \cite{IshiiMomose1991,JeonKim2007,Orlic2025}, and the
classifications of curves with infinitely many points of degree at most $4$
\cite{JeonKimSchweizer2020,Dalal2023,DerickxOrlic2026,DHJO}. Everything else is done by three methods,
each stated once:
\begin{enumerate}
\item[I.] \emph{Explicit pencils}: an explicit divisor $D$ with $\ell(D)\ge2$ gives $\gon_\Q\le\deg D$
 (\S\ref{sec:method-I}).
\item[II.] \emph{Finite-field exclusion}: at a prime of good reduction, a complete search over $\Fp$ shows
 that there is no function of degree at most $d$ (\S\ref{sec:method-II}). We use the gonality algorithm of
 Derickx and Terao \cite{DerickxTerao} and their package CurveArith. For the large curves $X_1(N)$ we keep
 their first phase (places and expansions of differentials) unchanged and replace their second phase by a
 search over a smaller family of divisors, run in compiled code (\S\ref{sec:phase2}). For $X_1(48)$ we
 search instead on the quotient by a diamond involution (\S\ref{sec:double-cover}).
\item[III.] \emph{Lifting arguments}: when a finite field allows more than $\Q$ does, a Mordell--Weil or
 local argument decides which classes over $\Fp$ come from $\Q$. This is needed for Theorems~A
 and~B and for one curve in Theorem~C.
\end{enumerate}

All computations were done in Magma~V2.29-4 \cite{Magma} and SageMath~10.7. The code, data and logs
that verify all computations in this paper can be found on
\begin{center}
 \url{https://github.com/marin-varivoda/intermediate-modular-curves};
\end{center}
\S\ref{sec:computations} describes how to rerun them.
\section{Preliminaries}\label{sec:prelim}

\subsection{Curves and models}
The groups $\Delta$ and the curves $X_\Delta(N)$ are enumerated with Derickx's package MD~Sage
\cite{MDSage} (\texttt{intermediate\_modular\_groups}), as in \cite{Orlic2025,DerickxOrlic2026}. We identify
each curve with its LMFDB label by checking that the corresponding open subgroup of $\GL_2(\Z/N\Z)$,
\[
 H_\Delta=\left\{\begin{pmatrix}a&b\\0&d\end{pmatrix}\in\GL_2(\Z/N\Z): a\in\Delta\right\},
\]
is conjugate to the group of the label. Rational Fourier expansions at the cusp $\infty$ describe the model
of $X_\Delta(N)$ that parametrizes $\mu_N$-markings; since $\Delta=\Delta^{-1}$, it is $\Q$-isomorphic to
the model parametrizing points of order $N$: quotienting by the marked subgroup and dualizing interchanges the two
moduli problems over $\Q$ and carries the diamond operator $\langle a\rangle$ to $\langle a^{-1}\rangle$, so it descends to
$X_\Delta(N)$ because $\Delta^{-1}=\Delta$. In particular the cusp
$\infty$ is a rational point of $X_\Delta(N)$.

For genus $g\ge 5$ we use the canonical model computed by MD~Sage
(\texttt{vanishing\_quadratic\_forms} with \texttt{minimize\_strategy="LLL"}); all lower bounds over
finite fields for Theorems~C and~D are computed on these models or on the models of $X_1(N)$ below (for
Theorem~A, on Magma's canonical models of $X_0(N)$). For $X_1(N)$ we use the plane models
of Sutherland and van Hoeij \cite{SutherlandX1}: each comes with explicit Tate normal form data $(E,P)$ over its
function field with $P$ of order $N$, which gives the birational map to $X_1(N)$ (compare \cite{Sutherland2012}).

\subsection{Good reduction}
All lower bounds over $\Q$ in this paper come from a prime of good reduction.

\begin{lemma}[Specialization]\label{lem:specialization}
Let $C/\Q$ be a smooth projective geometrically integral curve, and let $\mathcal C$ be a smooth proper model
of $C$ over $\Z_{(p)}$. Then $\gon_\Q C\ge\gon_{\Fp}\mathcal C_{\Fp}$. More generally, if $L$ is a line
bundle on $C$ with $h^0(L)\ge2$, then its extension to $\mathcal C$ has $h^0\ge2$ on $\mathcal C_{\Fp}$.
\end{lemma}
\begin{proof}
The second statement is upper semicontinuity of $h^0$ \cite[III.12.8]{Hartshorne1977}, applied to the
extension of $L$ (which exists because $\mathcal C$ is regular). If $f\colon C\to\PP^1$ has degree $d$, take
$L=f^*\mathcal O(1)$: its extension has degree $d$ and at least two sections on the special fibre, so
$\gon_{\Fp}\mathcal C_{\Fp}\le d$. This is standard; see for example \cite{Poonen2007} and \cite{NajmanOrlic2024}.
\end{proof}

We certify that a given model has good reduction at $p$ in one of two ways.
\begin{enumerate}
\item For the canonical model of an intermediate curve of genus $g$, we check that the reduction of the
 MD~Sage equations modulo $p$ is an integral curve with the Hilbert polynomial $(2g-2)t+1-g$ of the generic
 canonical curve, whose normalization has genus $g$, and which contains the reduction $(1:0:\dots:0)$ of the
 cusp $\infty$. Then the flat closure of $C$ in the integral model has the same special fibre (both have the same
 Hilbert polynomial and one contains the other); equality of arithmetic and geometric genus makes it normal,
 hence smooth, and the rational point makes it geometrically integral. (The primes $p\nmid N$ at which this can
 fail are controlled by Derickx and Najman \cite{DerickxNajman}; we only use primes where the check passes.)
\item For $X_1(N)$ with $p\nmid N$: the curve $X_1(N)$ has good reduction at $p$ \cite{DeligneRapoport1973}.
 Its function field is $\Q(x,y)$ with $F(x,y)=0$ the plane model of \cite{SutherlandX1}, scaled to a
 primitive polynomial in $\Z[x,y]$. We reduce $F$ modulo $p$ and check that $\deg_y\bar F=\deg_yF$, that
 $\bar F$ is irreducible over $\Fp(x)$, that $\Fp$ is algebraically closed in
 $K_p=\Fp(x)[y]/(\bar F)$, and that $K_p$ has genus $g(X_1(N))$. The field $K_p$ is the residue field of
 the Gauss valuation of $\Q(x,y)$ extending the $p$-adic valuation, so it is the function field of a
 component of the special fibre of some normal model of $X_1(N)$. Every such model is dominated by a
 blow-up of the smooth model, whose exceptional components are rational \cite[\S9.3, \S10.1]{Liu2002}; as
 $g(K_p)=g(X_1(N))\ge 2$, $K_p$ is the function field of $X_1(N)_{\Fp}$.
\end{enumerate}

\subsection{Tools from the geometry of curves}

\begin{lemma}[Castelnuovo--Severi {\cite[Theorem~3.11.3]{Stichtenoth2009}}]\label{lem:CS}
Let $C$ be a curve of genus $g$ over a perfect field, with morphisms $C\to C_1$ and $C\to C_2$ of degrees
$d_1,d_2$ to curves of genera $g_1,g_2$. If the two morphisms do not both factor through a common morphism
of degree $>1$, then $g\le d_1g_1+d_2g_2+(d_1-1)(d_2-1)$.
\end{lemma}

We use it mainly with $C_2=\PP^1$: a morphism $C\to\PP^1$ of degree $d$ that does not factor through a
morphism $C\to C_1$ of degree $m$ to a curve of genus $h$ satisfies $g\le mh+(m-1)(d-1)$. If $\gcd(d,m)=1$
the factorization condition holds automatically.

\begin{lemma}[Pullback]\label{lem:pullback}
If $\pi\colon C\to D$ has degree $m$ over $\Q$, then $\gon_\Q C\le m\cdot\gon_\Q D$.
\end{lemma}

\begin{lemma}[Hilbert irreducibility]\label{lem:hilbert}
If a curve $C/\Q$ has a morphism $C\to\PP^1$ of degree $d$ over $\Q$, then it has infinitely many points
of degree $d$.
\end{lemma}
\begin{proof}
By Hilbert's irreducibility theorem, the set of $t\in\PP^1(\Q)$ whose fibre is a single closed point of degree $d$
is the complement of a thin set, and in particular infinite.
\end{proof}

\begin{lemma}[{Frey \cite{Frey1994}, Abramovich \cite{Abramovich1996}}]\label{lem:frey}
If $C/\Q$ has infinitely many points of degree $d$, then $\gon_\Q C\le 2d$. For a modular curve of index
$\mu$ in $\mathrm{PSL}_2(\Z)$, $\gon_{\mathbb C}\ge\frac{\lambda_1}{24}\mu$ with $\lambda_1\ge\frac{975}{4096}$
by Kim and Sarnak \cite[Appendix~2]{KimSarnak2003}.
\end{lemma}

In particular, a modular curve with $119\mu>120000$ has $\gon_{\mathbb C}>10$ and so only finitely many
points of degree $5$; this is the form used in \cite[Theorem~3.16]{Orlic2025} and below.
\section{Three methods}\label{sec:methods}

Throughout, $C$ is a smooth projective geometrically integral curve of genus $g$ over a field $k$, and for a
divisor $D$ we write $L(D)=\{f\in k(C)^\times:\operatorname{div}f+D\ge0\}\cup\{0\}$ and $\ell(D)=\dim L(D)$.
A nonconstant $f\in k(C)$ has degree $\deg f=\deg(f)_\infty$, the degree of its pole divisor, and
$\gon_kC\le d$ if and only if there is an effective divisor $D$ of degree at most $d$ with $\ell(D)\ge2$.

\subsection{Method I: explicit pencils}\label{sec:method-I}
To prove $\gon_\Q C\le d$ we exhibit a rational effective divisor $D$ of degree $d$ and compute
$\ell(D)\ge2$ with Magma's Riemann--Roch routine. The divisors used are of three kinds:
\begin{enumerate}
\item divisors supported on rational cusps, or on cusps and other low-degree points, whose
 Riemann--Roch spaces we compute from $q$-expansions at all cusps, using Zywina's package
 \cite{ZywinaModular};
\item pullbacks along a forgetful map $X_\Delta(N)\to X_{\Delta'}(M)$ of a pencil of minimal degree on the
 target, whose gonality is known (\cref{lem:pullback});
\item low genus: if $g\ge2$ and $P\in C(\Q)$, then $\ell(K-(g-2)P)=\ell((g-2)P)+1\ge2$ by Riemann--Roch, so
 $\gon_\Q C\le g$; if $g\le1$ and $P\in C(\Q)$, then $\ell(2P)\ge2$ (and $\ell(P)=2$ if $g=0$).
\end{enumerate}
For $X_1(N)$ the upper bounds in Theorem~D are the degrees of the modular units of Derickx and van
Hoeij \cite[Table~1]{DerickxVanHoeij2014}, which also equal the degree in $y$ of the model of
\cite{SutherlandX1}.

\subsection{Method II: finite-field exclusion}\label{sec:method-II}
To prove $\gon_\Q C>d$, we choose a prime $p$ of good reduction (\cref{lem:specialization}) and show that
$C_{\Fp}$ has no nonconstant function of degree at most $d$. For this we use the algorithm of Derickx and Terao
\cite{DerickxTerao}, implemented in their Magma package CurveArith; it works over any finite field $\Fq$ (here
$q=p$) and has two phases.

\emph{Phase 1 (places and expansions).} Let $N_1=\#C(\Fq)$ and $n=\lceil N_1/(q+1)\rceil$. CurveArith
enumerates the places of $C_{\Fq}$ of degree at most $d-\min(n,d-1)$, fixes a basis $\omega_1,\dots,\omega_g$
of $H^0(C,\Omega^1)$, and at each place $P$ computes the first $m_P$ coefficients of the expansions of the
$\omega_i$ in a uniformizer at $P$, as vectors over $\Fq$ ($\deg P$ coordinates per coefficient). For an
effective divisor $D=\sum a_PP$ with $a_P\le m_P$ this gives a $g\times\deg D$ matrix $M_D$, whose columns
are the coefficients of index $<a_P$ at each $P$ in its support.

\begin{lemma}[{\cite{DerickxTerao}}]\label{lem:rank}
$\ell(D)=\deg D+1-\rank M_D$.
\end{lemma}
\begin{proof}
By Riemann--Roch, $\ell(D)=\deg D+1-g+\ell(K-D)$, and $\ell(K-D)=g-\rank M_D$ is the dimension of the space of
holomorphic differentials vanishing on $D$.
\end{proof}

\begin{lemma}[{\cite{DerickxTerao,DerickxVanHoeij2014}}]\label{lem:fibres}
If $C_{\Fq}$ has a nonconstant function $f$ of degree at most $d$, it has one whose pole divisor contains at
least $n$ distinct rational places.
\end{lemma}
\begin{proof}
The $N_1$ rational points are distributed among the $q+1$ fibres of $f$ over $\PP^1(\Fq)$, so one fibre
contains at least $n$ of them. Compose $f$ with an automorphism of $\PP^1$ over $\Fq$ moving that fibre to
$\infty$.
\end{proof}

\emph{Phase 2 (search).} CurveArith tests all effective divisors $D$ of degree $d$ containing at least $n$
rational places for $\ell(D)\ge2$ (\cref{lem:rank}). This is enough for every strict intermediate curve in
this paper. For the curves $X_1(N)$ with $41\le N\le50$ there are far too many such divisors, and we use
instead a smaller family of larger divisors, as follows.

\subsection{A smaller search family}\label{sec:phase2}
Let $S=\{P_1,\dots,P_{N_1}\}$ be the rational places, and also write $S=\sum P_i$. Fix an integer $b\ge1$.
For an effective divisor $Q$ with no rational place in its support and an effective divisor $E$ supported on
$S$, put $s=\#\supp E$ and
\begin{equation}\label{eq:budget}
 c(E,Q)=b\,s+\max(n-s,0)+\deg E+\deg Q.
\end{equation}
Let $\mathcal F_b(d)$ be the set of divisors $H=bS+E+Q$ with $\deg Q\le d-n$, $c(E,Q)\le d$, and $E$
maximal with this property for its $Q$ (for the order $E\le E'$).

\begin{proposition}\label{prop:covers}
Suppose $n\ge1$. If $C_{\Fq}$ has a nonconstant function of degree at most $d$, then there is
$H\in\mathcal F_b(d)$ such that $L(H)$ contains a nonconstant function of degree at most $d$.
\end{proposition}
\begin{proof}
By \cref{lem:fibres} there is such an $f$ whose pole divisor $D=A+Q$ has at least $n$ distinct rational
places in its support, where $A$ is supported on $S$ and $Q$ on the other places. Put
$E=\sum_{P\in S}\max(v_P(A)-b,0)P$ and $s=\#\supp E$. A place in $\supp E$ contributes $b+v_P(E)$ to $\deg A$,
and $\supp A$ has at least $\max(n-s,0)$ further places, each contributing at least $1$. Hence
$c(E,Q)\le\deg A+\deg Q=\deg f\le d$, and in particular $\deg Q\le d-n$. Since $v_P(A)\le b+v_P(E)$ for every
$P\in S$, we have $D\le bS+E+Q$. Enlarging $E$ to a maximal $E'$ with $c(E',Q)\le d$ keeps
$D\le bS+E'+Q$, so $f\in L(bS+E'+Q)$ with $bS+E'+Q\in\mathcal F_b(d)$.
\end{proof}

The divisors in $\mathcal F_b(d)$ have degree larger than $d$, so $\ell(H)\ge2$ is not enough: we need the
least degree of a nonconstant function in $L(H)$. It can be read off from the same matrix $M_H$.

\begin{lemma}\label{lem:kernel}
Let $H$ be effective. The space $L(H)/k$ is isomorphic to the space of vectors $x$ with $M_Hx=0$. Under this
isomorphism, the order of the pole of $f$ at $P\in\supp H$ is the largest $j\le v_P(H)$ such that the
coordinates of $x$ belonging to the coefficient of index $j-1$ at $P$ are not all zero.
\end{lemma}
\begin{proof}
For each $P\in\supp H$ let $V_P=\mathcal O_P/\mathfrak m_P^{v_P(H)}$ and $V=\bigoplus V_P$. Write each $\omega_i$ as $g_i\,du_P$ with $u_P$ the uniformizer chosen by CurveArith at $P$; the columns of
$M_H$ are the coordinates (over $k$, using a $k$-basis of the residue field at $P$) of the jets of the $g_i$ in
$V$, in which, for every $j$, the subspace $\mathfrak m_P^j/\mathfrak m_P^{v_P(H)}$
is cut out by the vanishing of the coordinates of index $<j$; indeed, CurveArith computes the coefficients
one at a time, subtracting a lift of each coefficient and dividing by the uniformizer, and this is
$k$-linear. The pairing $(\phi,\eta)\mapsto\sum_P\operatorname{Tr}_{k(P)/k}\Res_P(\phi\,\eta\,du_P)$ identifies the principal parts $\bigoplus_P\mathfrak m_P^{-v_P(H)}/\mathcal O_P$
with the dual of $V$, a principal part having pole order $\le j$ at $P$ exactly when it annihilates
$\mathfrak m_P^{j}/\mathfrak m_P^{v_P(H)}$. By the residue theorem and Serre duality, a principal part is
the principal part of a function in $L(H)$ (unique up to constants) if and only if it annihilates the image
of $H^0(C,\Omega^1)$ in $V$, that is, if and only if its coordinate vector $x$ satisfies $M_Hx=0$.
\end{proof}

\emph{Precision.} The expansions computed in Phase~1 suffice for every $H\in\mathcal F_b(d)$: at a rational place
$P\in\supp E$ we have $v_P(H)=b+v_P(E)\le bs+\deg E\le d$, which is CurveArith's precision there, and at a place
$P$ of degree $e\ge2$ we have $e\,v_P(H)\le\deg Q\le d-n$, so $v_P(H)\le\lfloor(d-\min(n,d-1))/e\rfloor$, which is
CurveArith's precision there. (The program checks this before searching.)

So for $H\in\mathcal F_b(d)$ we compute the kernel of $M_H$ and check whether some nonzero kernel vector, up
to scalars, gives a pole divisor of degree at most $d$. By \cref{prop:covers}, if this fails for every
$H\in\mathcal F_b(d)$, then $\gon_{\Fq}C>d$. The same argument with $b=0$ and all effective divisors of degree
at most $d$ handles curves without rational points; we did not need this case.

\begin{remark}
\Cref{prop:covers} and \cref{lem:kernel} are an elementary variation on the second phase of
\cite{DerickxTerao}; the first phase is used unchanged. The point is that $\mathcal F_b(d)$ is much smaller
than the set of divisors of degree $d$ with $n$ rational places: the rational part is fixed to $bS$ up to the
small excess $E$, and the kernels of $M_H$ have dimension at most $2$ or $3$ in all our computations.
For a fixed $Q$, the divisors $bS+E+Q$ share the block of rows of $bS+Q$, so we eliminate those once and
add the rows of $E$ incrementally. This variation was first implemented as an option
\texttt{Al:="Cover"} of CurveArith's \texttt{CAHasFunctionOfDegreeAtMost}; for the curves $X_1(N)$ we run
the same search in a short C++ program (\S\ref{sec:computations}).
\end{remark}

\subsection{Method III: lifting}\label{sec:method-III}
When $C_{\Fp}$ has more functions or divisor classes of a given kind than can come from $\Q$, we use the
structure of $J(\Q)$, where $J=\Jac C$, to decide which of them lift. The arguments are of three types, each
explained where it is used:
\begin{enumerate}
\item \emph{rank zero}: if $J(\Q)$ is finite, reduction at an odd good prime is injective on $J(\Q)$, and
 a finite list of classes over $\Fp$ can be compared with explicit rational classes
 (\S\ref{sec:quintic-x0}, \S\ref{sec:quintic-intermediate});
\item \emph{a known positive-rank part}: if $J(\Q)=A(\Q)+T$ for an explicit abelian subvariety $A$ and an
 explicit finite group $T$, the reduction of $J(\Q)$ is contained in an explicit subgroup of $J(\Fp)$
 (the curve $43.132.9.a.2$ in \S\ref{sec:gonality-intermediate}, and Route Q in \S\ref{sec:quintic-x0}, \cref{tab:routeQ});
\item \emph{local rigidity}: if over $\Fp$ the relevant locus agrees, to first order, with a known family
 defined over $\Q$, a formal implicit function argument shows that it agrees over $\Z_p$
 (\cref{lem:local}).
\end{enumerate}
\section{Gonality of intermediate curves: proof of Theorem C}\label{sec:gonality-intermediate}

There are $76$ strict intermediate curves with $N\le50$, of which $52$ have $N\le40$; for the $39$ of these of genus $\ge3$ both bounds
are in \cite[Theorem~1.2, Table~1]{Orlic2025}, the other $13$ have genus $\le1$, and we only recall them in \cref{tab:C}. The remaining $24$
curves, with $41\le N\le50$, are treated as follows.

\subsection{Upper bounds}
Each upper bound is Method~I.
\begin{itemize}
\item \emph{Pullbacks.} Fourteen curves map, by a forgetful map of degree $m$, to a curve whose gonality $\gamma$
 is known, with $m\gamma$ equal to the lower bound. The targets have genus at most $3$, or are hyperelliptic,
 or are curves $X_1(M)$ with $M\le 40$ \cite{DerickxVanHoeij2014}; the degree and target of each map are listed in
the data file of the accompanying repository.
\item \emph{Cuspidal divisors.} For five curves ($41.168.11.b.1$, $41.210.11.b.1$, $41.420.21.d.1$,
 $45.432.21.g.2$, $49.392.19.i.2$), a rational divisor $D$ supported on the cusps has degree equal to the
 lower bound and $\ell(D)\ge2$. We compute $\ell(D)=\deg D+1-g+\ell(K-D)$, where $H^0(K-D)$ is the space of
 weight-$2$ cusp forms vanishing to the prescribed orders at the cusps, from $q$-expansions at all cusps
 computed with Zywina's package \cite{ZywinaModular}. Conditions at one cusp of each Galois orbit suffice,
 since the basis is rational.
\item \emph{Published.} $44.144.8.c.4$ is in Orli\'c's list of $\Q$-pentagonal curves
 \cite[Theorems~1.4--1.5]{Orlic2025}, and $45.144.5.a.1$ in the list of $\Q$-tetragonal curves
 \cite[Theorem~1.3]{Orlic2025}; $49.168.3.b.1$ is a nonhyperelliptic genus-$3$
 curve with a rational point.
\item \emph{Two special divisors at level $43$.} For $43.308.15.b.1$ (genus $15$) we take $D=3C_3+C_4+P_5$, where
 $C_3,C_4$ are rational cusps and $P_5$ is a point of degree $5$, the image of a point of degree $15$ on
 $X_1(43)$ found by van Hoeij \cite{vanHoeij2012}; then $\deg D=9$ and $\ell(D)=2$. For $43.132.9.a.2$
 (genus $9$), let $\pi\colon X\to X_0^+(43)=E$ be the map of degree $6$ to the elliptic curve $43a1$; with rational
 cusps $C_1,C_2,C_3$, the divisor $D=3C_3+7C_2-3C_1+\pi^*(G)-\pi^*(O)$ for a point $G\in E(\Q)$ of infinite
 order has degree $7$ and $\ell(D)=2$.
\end{itemize}

\subsection{Lower bounds}
Twenty-one of the $24$ lower bounds are Method~II, using CurveArith's \texttt{CAGonality} unchanged at a prime
of good reduction (\cref{tab:C}). Six of these curves ($41.84.5.a.1$, $44.144.8.c.4$, $45.144.5.a.1$ and the
three curves of genus $5$ and $7$ at level $48$) are also covered by Orli\'c's classification
\cite{Orlic2025}; we recompute them as a check. The longest computation, for $48.384.19.bc.1$ at $p=7$, takes
about an hour; all others take at most eleven minutes. The curves $49.168.3.b.1$ and $50.180.4.a.2$ have
genus $3$ and $4$ and are not hyperelliptic \cite{IshiiMomose1991}, hence have gonality at least $3$.

The remaining curve, $43.132.9.a.2$, needs Method~III.

\subsection{The curve \texorpdfstring{$43.132.9.a.2$}{43.132.9.a.2}}
Let $X=X_\Delta(43)$ with $\Delta=\langle-1,2\rangle=\{\pm1,\pm2,\pm4,\pm8,\pm11,\pm16,\pm21\}$. It has genus
$9$, three rational cusps $C_1,C_2,C_3$ and one cusp orbit $C_{(3)}$ of degree $3$. The forgetful map
$X\to X_0(43)$ has degree $3$, and composing with the quotient by the Atkin--Lehner involution gives
$\pi\colon X\to X_0^+(43)=E$, of degree $6$, where $E$ is the elliptic curve $43a1$ of rank $1$.

\begin{proposition}\label{prop:43}
$\gon_\Q X=7$.
\end{proposition}

The Castelnuovo--Severi inequality does not help here: the maps to $X_0(43)$ (genus $3$, degree $3$) and to
$E$ (degree $6$) exclude nothing in degrees $2$ to $6$. Over $\F_2$ the gonality is $5$ and over $\F_5$ it is
$4$; we use $p=3$, where $X$ has good reduction.

\begin{proof}
The upper bound is the divisor above. For the lower bound, let $J=\Jac X$, and let $c=C_3$ be the base point of
the Abel--Jacobi map.

\emph{Step 1: over $\F_3$.} \texttt{CAGonality} gives $\gon_{\F_3}X_{\F_3}=6$, so by \cref{lem:specialization}
there is no function of degree at most $5$ over $\Q$. Here $n=\lceil 4/4\rceil=1$, and testing the $845$
effective divisors of degree $6$ over $\F_3$ containing a rational place shows that exactly four have
$\ell\ge2$, and that they are linearly equivalent. So $W^1_6(X_{\F_3})(\F_3)$ is a single class $T$, and
$T-6c$ has order $1596$ in $J(\F_3)$, a group of order $19152$.

\emph{Step 2: the rational points of $J$.} Let $A=\pi^*E\subseteq J$ and $B=J/A$. The pullback $\pi^*$ is
injective: a nontrivial kernel would make $\pi$ factor through an isogeny onto $E$ of degree dividing $6$, but
$43a1$ has no rational cyclic isogeny and $E(\Q)_{\mathrm{tors}}=0$. So $A(\Q)\cong E(\Q)\cong\Z$. The
abelian variety $B$ is isogenous to a product of modular abelian varieties of dimensions $2$, $2$ and $4$
whose $L$-values at $1$ are nonzero (exact modular-symbol computations of $L$-ratios $2/7$, $3/16$,
$1/361$); by Kato \cite[Corollary~14.3]{Kato2004}, $B(\Q)$ is finite. Its order divides
$\gcd_{p}\#B(\F_p)$ over odd primes $p\ne43$, since reduction is injective on rational torsion at such
primes \cite[Appendix]{Katz1981}; the gcd over $p=3,5,23$ is already $532$. (Point counts are invariant under
isogeny, so we may compute them on the modular abelian variety isogenous to $B$.)
On the other hand, the subgroup $\mathcal C\subseteq J(\Q)$ generated by $[C_2-C_1]$, $[C_3-C_1]$ and
$[C_{(3)}-3C_3]$ is finite (its generators are torsion, as checked below; this is also an instance of the theorem of
Manin and Drinfeld \cite{Manin1972,Drinfeld1973}), of order at least $532$ (its reduction modulo $3$ has order
$532$), and it meets $A(\Q)$ trivially, as $A(\Q)$ is torsion-free. So $\mathcal C$ maps isomorphically
onto $B(\Q)$, and
\[
 J(\Q)=A(\Q)+\mathcal C.
\]

\emph{Step 3: conclusion.} Let $H_3\subseteq J(\F_3)$ be the subgroup generated by $\pi^*E(\F_3)$ and the
reduction of $\mathcal C$. It has order $3192$, and $T-6c\notin H_3$. If $f\colon X\to\PP^1$ had degree $6$
over $\Q$, the class of $L=f^*\mathcal O(1)$ would satisfy $[L-6c]\in J(\Q)$, so its reduction would lie in
$H_3$; but by \cref{lem:specialization} the reduction of $L$ is a degree-$6$ class with two sections, that is
$T$. This contradiction shows $\gon_\Q X\ge7$.
\end{proof}

\begin{remark}
The saturation in Step~2 cannot be avoided: $J(\F_3)$ has cyclic $3$-part, and the $3$-component of $T-6c$
lies in $\pi^*E(\F_3)$, so $T-6c\notin H_3$ only because of its $2$-, $7$- and $19$-parts. In particular the
coarser statement ``$532\cdot J(\Q)\subseteq A(\Q)$'' would not suffice.
\end{remark}

The certificate for \cref{prop:43} runs in about $12$ seconds; the computations of $L$-ratios and of
$\#B(\F_p)$ take a few minutes.
The scripts work with the LMFDB model of $X$, its map to $X_0(43)$ and its cusps; the script
\texttt{identify\_model.m} shows that this model is isomorphic over $\Q$ to the MD~Sage model, by an explicit
linear change of coordinates (found by matching points over $\F_{101}$ and $\F_{103}$, and checked exactly). The three
classes generating $\mathcal C$ are certified to be torsion directly over $\Q$ (\texttt{torsion\_classes.m}): with
$D_1=C_2-C_1$, $D_2=C_3-C_1$ and $E=C_{(3)}-(C_1+C_2+C_3)$, the divisors $38D_1$, $38D_2$ and $7E$ are principal,
and $C_{(3)}-3C_3=E+D_1-2D_2$. So the proof does not depend on knowing that these points are cusps.
\section{Gonality of \texorpdfstring{$X_1(N)$}{X1(N)}: proof of Theorem D}\label{sec:gonality-x1}

\begin{proof}[Proof of Theorem D]
\emph{Upper bounds.} Derickx and van Hoeij \cite[Table~1]{DerickxVanHoeij2014} give, for each $N$, a modular unit
on $X_1(N)$ over $\Q$ whose degree is the value in Theorem~D; it is also the degree in $y$ of the plane model of
\cite{SutherlandX1}, i.e.\ the degree of the coordinate function $x$.

\emph{Lower bounds.} Let $u$ be the value in Theorem~D and $d=u-1$. For $N=48$ see \S\ref{sec:double-cover}. For the other $N$ we apply Method~II to $X_1(N)$ at the prime
$p$ of \cref{tab:D}, which does not divide $N$; \S\ref{sec:prelim} explains why the reduction of the plane model
of \cite{SutherlandX1} computes $X_1(N)_{\Fp}$, and we check its genus. Phase~1 is CurveArith's: the places of
degree at most $R=d-\min(n,d-1)$ and the expansions of a basis of holomorphic differentials at them, with
CurveArith's precisions. Phase~2 is the search of \S\ref{sec:phase2} over the family $\mathcal F_b(d)$. In every
case it finds no function of degree at most $d$. By \cref{prop:covers} and \cref{lem:kernel},
$\gon_{\Fp}X_1(N)_{\Fp}>d$, and by \cref{lem:specialization}, $\gon_\Q X_1(N)\ge u$.
\end{proof}

\begin{table}[ht]\centering\small
\caption{Theorem D: Method II for $X_1(N)$. $n=\lceil N_1/(p+1)\rceil$; places up to degree $R=d-\min(n,d-1)$;
 $b$ the baseline; $\#\mathcal F_b(d)$ the number of divisors searched. Time: Phase 1 (its longest process, on a shared machine) + Phase 2.}\label{tab:D}
\begin{tabular}{rrrrrrrrrrr}\toprule
$N$ & $g$ & $p$ & $d$ & $N_1$ & $n$ & $R$ & places & $b$ & $\#\mathcal F_b(d)$ & time\\\midrule
41 & 51 & 2 & 21 & 20 & 7 & 14 & 2669 & 2 & 2643150 & 171 min + 14 s\\
42 & 25 & 5 & 11 & 12 & 2 & 9 & 277397 & 1 & 3860784 & 648 min + 153 s\\
43 & 57 & 2 & 23 & 21 & 7 & 16 & 8515 & 2 & 17156720 & 367 min + 50 s\\
44 & 36 & 3 & 14 & 15 & 4 & 10 & 9518 & 1 & 3133892 & 49 min + 25 s\\
45 & 41 & 2 & 17 & 12 & 4 & 13 & 1608 & 2 & 265608 & 86 min + 1 s\\
46 & 45 & 3 & 18 & 22 & 6 & 12 & 69995 & 1 & 374417725 & 884 min + 950 s\\
47 & 70 & 2 & 28 & 23 & 8 & 20 & 110354 & 2 & 651609964 & 1813 min + 3565 s\\
49 & 69 & 2 & 20 & 21 & 7 & 13 & 1497 & 2 & 2167431 & 207 min + 13 s\\
50 & 48 & 3 & 14 & 20 & 5 & 9 & 3369 & 2 & 41835 & 161 min + 1 s\\
\bottomrule\end{tabular}\end{table}

The second phase also counts the divisors it tests and compares the total with an independent count of
$\#\mathcal F_b(d)$ computed from the numbers of places of each degree; they agree in every case.
As a check of the whole pipeline, the same programs reproduce $\gon_{\F_2}X_1(29)=11$ (degree $10$ is
excluded after testing $8568$ divisors, and a function of degree $11$ is found). For $X_1(47)$, an independent earlier computation,
with a different implementation of the first phase (a different enumeration of places and expansions), tested
exactly the same number $651{,}609{,}964$ of divisors and also found no function of degree at most $28$.

\subsection{The level \texorpdfstring{$48$}{48}: a double cover}\label{sec:double-cover}
For $N=48$ Method~II over $\F_5$ or $\F_7$ would need all places of degree up to $12$, which is out of reach.
We use instead the quotient of $X=X_1(48)$ by the diamond operator $\langle 23\rangle$, an involution; the
quotient is $Y=X_{\{\pm1,\pm23\}}(48)$, of genus $13$, and $X$ has genus $37$. Everything is computed on $Y$.

Let $k$ be a finite field of odd characteristic, and $\pi\colon X\to Y$ a morphism of degree $2$ of smooth
curves over $k$, the quotient by an involution $\sigma$. For a function $a$ on $Y$ write $\pi^*a=a\circ\pi$;
these are exactly the $\sigma$-invariant functions on $X$. Fix a function $w$ on $X$ with $\sigma w=-w$; then
$w^2=\pi^*\delta$ for a function $\delta$ on $Y$. Write $\operatorname{div}_Y\delta=B-2H$ with $B$ the sum of
the branch points, so that $c:=\deg H=\tfrac12\deg B=g_X-2g_Y+1$. (For a hyperelliptic curve
$w^2=\delta(x)$ over $Y=\PP^1$, all of this is classical: every function is $a(x)+b(x)w$.)

\begin{lemma}\label{lem:dc}
\begin{enumerate}
\item Every function $f$ on $X$ is uniquely $f=\pi^*a+\pi^*b\cdot w$ with $a,b$ functions on $Y$, and $f$ comes
 from $Y$ if and only if $b=0$.
\item Put $t=2a$ and $n=a^2-b^2\delta$. Then $f+\sigma f=\pi^*t$, $f\cdot\sigma f=\pi^*n$, and for every place
 $P$ of $Y$ the total pole order of $f$ above $P$ is $\deg P\cdot\max(0,-v_P(t),-v_P(n))$. In particular
 $\deg_Xf\ge\deg_Yn$ and $\deg_Xf\ge\deg_Yt$.
\item If $\deg_Xf=d$ and $b\ne0$, there are an effective divisor $E$ on $Y$ of degree $d-c$ and $u\ne0$, $h$ in
 $L_Y(E+H)$ with $f=(\pi^*h+w)/\pi^*u$.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) Take $\pi^*a=(f+\sigma f)/2$ and $\pi^*b=(f-\sigma f)/(2w)$, both $\sigma$-invariant.
(2) The first two identities are direct. Measure the orders of the two values of $f$ above $P$ in units of
$v_P$, say $\alpha,\beta$ (if a single ramified point $Q$ lies above $P$, then $\alpha=\beta=v_Q(f)/2$; if a single
unramified point $Q$ of degree $2\deg P$ lies above $P$, then $\alpha=\beta=v_Q(f)$). The
pole order above $P$ is $\deg P(\max(0,-\alpha)+\max(0,-\beta))$, while $v_P(n)=\alpha+\beta$ and
$v_P(t)\ge\min(\alpha,\beta)$ with equality if $\alpha\ne\beta$. If $\alpha,\beta\ge0$ both sides vanish; if
$\alpha<0\le\beta$ then $v_P(t)=\alpha\le v_P(n)$ and both sides are $-\alpha$; if $\alpha,\beta<0$ then
$v_P(n)<v_P(t)$ and both sides are $-\alpha-\beta$.
(3) Since $\operatorname{div}_Xw=R-\pi^*H$ with $R$ the ramification divisor, for every divisor $A$ on $Y$ the
poles of $f$ are bounded by $\pi^*A$ if and only if $a\in L_Y(A)$ and $b\in L_Y(A-H)$. Take $A=\pi_*$ of the pole
divisor of $f$, of degree $d$, and $E=\operatorname{div}b+A-H\ge0$, $u=1/b$, $h=a/b$.
\end{proof}

\begin{proposition}\label{prop:dc}
$X$ has no nonconstant function of degree at most $m$ if $2\gon_k Y>m$ and, for every effective divisor $E$ on $Y$
of degree $m-c$, every $u\ne0$ and every $h$ in $L_Y(E+H)$, the function $(\pi^*h+w)/\pi^*u$ has degree $>m$.
\end{proposition}
\begin{proof}
Functions with $b=0$ are $\pi^*a$, of degree $2\deg_Ya$. For the others use \cref{lem:dc}(3) and enlarge $E$
by $m-d$ points, which only enlarges $L_Y(E+H)$.
\end{proof}

For $X_1(48)$ we take $k=\F_7$ and $m=15$, so $c=12$ and $E$ has degree $3$. The diamond operator $\langle23\rangle$
is defined over $\Q$ and extends to the smooth model of $X_1(48)$ over $\Z_{(7)}$; since $2$ is invertible, the
quotient commutes with base change, so $Y_{\F_7}=X_1(48)_{\F_7}/\langle23\rangle$ is the reduction of $Y$. We use
MD~Sage's canonical model of $Y$ (coordinates $y_0,\dots,y_{12}$, a basis of $S_2(\Gamma_Y)$ with integral
$q$-expansions, $y_0=q+O(q^2)$), checked to have good reduction at $7$ as in \S\ref{sec:prelim}. Let
$\psi\in S_2(\Gamma_1(48))$ have integral $q$-expansion, nonzero modulo $7$, with $\langle23\rangle\psi=-\psi$, and put
$w=\psi/y_0$, a function on $X_1(48)$ with $\sigma w=-w$. Then $\delta=w^2=\psi^2/y_0^2$ is a ratio of two forms of
weight $8$ on $Y$, $\delta=\mathrm{Nm}(y)/\mathrm{Dn}(y)$, from an identity $\psi^2\,\mathrm{Dn}(y)=y_0^2\,\mathrm{Nm}(y)$
with integral coefficients checked to the Sturm bound. Reducing modulo $7$, $\bar w$ is a function on
$X_1(48)_{\F_7}$ with $\sigma\bar w=-\bar w$ and $\bar w^2=\bar\delta$, so the function field of $X_1(48)_{\F_7}$ is
$\F_7(Y)(\sqrt{\bar\delta})$; as a check, Magma computes its genus as $37$. \texttt{CAGonality} gives
$\gon_{\F_7}Y=8$, so $2\gon Y=16>15$. There are $1328$ effective divisors $E$ of degree $3$ on $Y_{\F_7}$; each
$L_Y(E+H)$ has dimension $3$, giving $57$ choices of $u$ up to scalars and $49$ of $h$ modulo $u$, and
$3{,}709{,}104$ functions in all. The bound $\deg_Xf\ge\deg_Yn$ of \cref{lem:dc}(2) excludes all but $18$ of
them, and for those the exact degree, computed on $Y$, is at least $16$. Hence $\gon_{\F_7}X_1(48)>15$, and
$\gon_\Q X_1(48)=16$ by \cref{lem:specialization}. The divisors $E$ are split into parts by a key that depends only on $E$ (not on the order in which Magma lists
places), so the parts cover every $E$ exactly once; the longest part took $12.5$ hours. The
degree formula of \cref{lem:dc}(2) was checked against Magma's degree computation on $X$ itself for a sample of
the candidates, and the whole procedure was tested on $X_1(24)$ and $X_1(26)$, where it reproduces the known
gonalities.
\section{Points of degree 5 on \texorpdfstring{$X_0(N)$}{X0(N)}: proof of Theorem A}\label{sec:quintic-x0}

Let $C=X_0(N)$ for one of the thirty levels of Theorem~A, of genus $g$, and $J=\Jac C$. The rational cusp
$\infty$ identifies $\Pic^5(C)$ with $J$. For $r\ge0$ let $W^r_5\subseteq\Pic^5(C)$ be the locus of classes
with $h^0\ge r+1$. \Cref{tab:A} lists, for every level, the argument for each of the two assertions of
Theorem~A.

\subsection{The framework}
\begin{lemma}\label{lem:inject}
If $C$ has no function of degree $5$ over $\Q$, then the points of degree $5$ inject into $\W(\Q)$, and a
point of degree $5$ is the unique effective divisor in its class.
\end{lemma}
\begin{proof}
If $D$ is a point of degree $5$ and $h^0(D)\ge2$, a nonconstant $f\in L(D)$ over $\Q$ has a nonzero rational
pole divisor bounded by $D$; since $D$ is irreducible, it equals $D$, so $\deg f=5$.
\end{proof}

Up to isogeny, $J$ is a product of modular abelian varieties. Let $\A\subseteq J$ be the sum of the isotypic
components whose $L$-function vanishes at $s=1$; the vanishing is decided by exact modular-symbol
computations. By Kolyvagin and Logachev \cite{KolyvaginLogachev1989}, $(J/\A)(\Q)$ is finite. By Faltings
\cite{Faltings1991}, the Zariski closure of $\W(\Q)$ is a finite union of translates of abelian subvarieties;
a positive-dimensional component with dense rational points has its direction in $\A$. So it suffices to show,
in addition to excluding functions of degree $5$, that every positive-dimensional translate $T\subseteq\W$
with $T(\Q)$ dense has a fixed proper subdivisor on its locus with $h^0=1$: then only finitely many classes
can contain an irreducible divisor.

At each working prime $p\nmid N$ we check that the reduction of Magma's canonical model is an integral curve
of arithmetic genus $g$, with normalization of genus $g$ and a rational point (\S\ref{sec:prelim}), and we
enumerate all effective divisors of degree $5$ over $\Fp$, for every partition of $5$ into degrees of places.
If $x+B\subseteq\W$ over $\Q$, then the reduction contains the entire coset $\bar x+B(\Fp)$ in $\W(\Fp)$.

\begin{lemma}[Local rigidity]\label{lem:local}
Let $R=\Z_p$, let $Z$ be a closed subscheme of a scheme smooth over $R$, and let $U\to Z$ be a morphism from a
smooth $R$-scheme of relative dimension $n$. Let $u\mapsto z$ be rational points of the special fibres. If
$\dim T_zZ_{\Fp}=n$ and $T_uU_{\Fp}\to T_zZ_{\Fp}$ is an isomorphism, then $U\to Z$ induces an isomorphism of
completed local rings at $u$ and $z$.
\end{lemma}
\begin{proof}
Lifting cotangent generators gives a surjection $R[[x_1,\dots,x_n]]\to\widehat{\mathcal O}_{Z,z}$; its
composite with $\widehat{\mathcal O}_{Z,z}\to\widehat{\mathcal O}_{U,u}\cong R[[t_1,\dots,t_n]]$ has invertible
linear part modulo $p$, so it is an isomorphism, and the surjection is injective.
\end{proof}
We apply this to $Z_S=\bigcap_{s\in S}(\W-s)$ for a finite subgroup $S\subseteq J(\Fp)$, after lifting $S$
to $J(\Z_p)$: prime-to-$p$ torsion points lift uniquely to torsion points over $\Z_p$ (and if $S$ lies in the
reduction of both a candidate direction and a known abelian subvariety $B$, the lifts lie in both); when the
direction is already known to be $B$, we may lift points of $B(\Fp)$ arbitrarily to $B(\Z_p)$, including
$p$-primary points, since both families are stable under all of $B$. At a class with $h^0(D)=1$ the annihilator
of $T_{[D]}\W$ in $H^0(K_C)$ is $H^0(K_C-D)$, and at a class with $h^0\ge2$ the tangent space is everything. A known family $V\subseteq\W$ defined over $\Q$
which fills the local branch then contains every candidate translate through that residue class. (For the
tangent space of $W^1_5$ and the base-point-free pencil trick used below see \cite[Chapter~IV]{ACGH1985}.)

\subsection{Excluding functions of degree 5}
\emph{The double-cover inequality.} If $C$ has an involution $w$ with quotient of genus $h$ and $g>2h+4$, a
morphism of degree $5$ cannot exist, even over $\overline\Q$, by \cref{lem:CS} (the degrees $2$ and $5$ are
coprime). This applies to $17$ levels (\cref{tab:A}). For $12$ of them, namely
$83,89,92,101,107,111,131,141,142,143,167,191$, these levels are already excluded in
\cite[Proposition~5.2]{DHJO} (for $142,143,167,191$ via a map of degree $4$ to $\PP^1$); it is new for $118,123,145,155,159$.

\emph{Published gonality.} $\gon_\Q X_0(82)=6$ \cite[Theorem~1.4]{NajmanOrlic2024}.

\emph{Empty special loci.} At $N=86,99,128$ and $p=7,5,3$ respectively, the numbers of effective divisors of
degree $5$ over $\Fp$ and of their classes are equal, so $W^1_5(\Fp)=\varnothing$, and \cref{lem:specialization}
applies.

\emph{Local pencils.} At $N=77,85,88$ the curve has $\Q$-gonality $4$ \cite[Theorem~1.2]{NajmanOrlic2024}. At
$p=5,7,3$, every special divisor of degree $5$ has $h^0=2$ and is $A_4+P$ for the unique class $A_4$ of degree $4$
with two sections, and $\dim T_{[D]}W^1_5=8-g+h^0(K-D-A_4)=1$ (the base-point-free pencil trick). The rational
tetragonal pencil reduces to a class of degree $4$ with two sections, which must be $A_4$ (an extra section would
give a class of degree $5$ with $h^0\ge3$ after adding a rational point); so it lifts $A_4$, the Abel map $P\mapsto A_4+P$ is an immersion, and \cref{lem:local} identifies
$W^1_5$ near every relevant point with this family. Its sections come from $A_4$, so a rational class in
$W^1_5$ has a base point. Level $74$ uses the same argument at the two primes $3$ and $5$. The possible positive-rank part $A$ is isogenous
to $E_{37}^2$, and the rank-zero quotient $R=J/A$ has $\#R(\Q)$ dividing $513$ (the gcd of its point counts at
$q\le31$). Since $\#E_{37}(\F_3)=7$ and $\#E_{37}(\F_5)=8$, $A(\F_3)$ is a $7$-group and $A(\F_5)$ a $2$-group, so
$A(\Q)_{\mathrm{tors}}$, which injects into both, is trivial. The four rational cusps generate a torsion group
$\mathcal C$ whose reductions modulo $3$ and $5$ have order $513$; it meets the torsion-free $A(\Q)$ trivially, so it
maps onto $R(\Q)$, and every rational class lies in $\mathcal C+A(\Q)$, whose reduction lies in
$\bar{\mathcal C}+A(\Fp)\subseteq\bar{\mathcal C}+J(\Fp)[\ell^\infty]$ with $\ell=7,2$. At
$p=3$ (using $J(\F_3)[7^\infty]$, of order $49$) and at $p=5$ (using $J(\F_5)[2^\infty]$, of order $256$) the
local pencil argument applies outside two exceptional cusp labels each, and the exceptional labels at the two
primes are disjoint; so every rational class is covered at one of the primes.

The remaining levels are treated with the finiteness argument below: rank-zero lifting at $96,98,100$ and the
quotient sieve at $79,103$.

\subsection{Route R: direct reduction}
\emph{Rank zero ($96,98,100$).} Here $L(J,1)\ne0$, so $J(\Q)$ is finite and reduction is injective on it at
the odd good primes $p=7,3,3$. This is the situation of \cite[Proposition~5.5]{NajmanVarivoda}. The computation
shows that every special class of degree $5$ over $\Fp$ has $h^0=2$ and is $\bar A_4+\bar P$ for a class $\bar
A_4$ of degree $4$ with two sections and a point $\bar P\in C(\Fp)$, and it lifts every such $\bar A_4$ to a
rational class $A_4$. At $96$ and $100$ every point of $C(\Fp)$ is the reduction of a rational point $P$, and
injectivity forces a rational special class to be $A_4+P$, which has the base point $P$. At $98$ only four of the
six points of $C(\F_3)$ (the cusps) lift. If $L$ is a rational special class, then $L-A_4$ is a rational class of
degree $1$ whose reduction is $\bar P$; its push-forward along the degeneracy map $X_0(98)\to X_0(49)=E$ is a point
of $E(\Q)\cong\Z/2$ reducing to the image of $\bar P$. For a noncuspidal $\bar P$ that image is a noncuspidal
point of $E(\F_3)\cong\Z/4$ (degeneracy maps send noncusps to noncusps), of order $4$, which is not the
reduction of a point of $E(\Q)$. So $\bar P$ is a cusp, and $L=A_4+P$ with a fixed rational cusp $P$.

\emph{Empty cosets ($85,88,92$).} Here $\A$ is simple, of dimension $2,1,1$. At $p=3$, the $\ell$-primary
subgroup of $\A(\F_3)$, for $\ell=17,7,7$, has no full coset inside $\W(\F_3)$; so no translate exists.

\subsection{Route Q: a rank-zero quotient (nine prime levels)}
For $N\in\{79,83,89,\allowbreak 101,103,107,\allowbreak 131,167,191\}$ let $Y=C/w_N$, $A=\pi^*\Jac Y$ and $R=J/A$; here $\A=A$ is simple
of multiplicity one and $R$ has rank $0$. With $t=[w_N(\infty)-\infty]$, the gcd $m$ of $\#R(\F_q)$ over
$q\le19$ odd equals the order of $\bar t$ in $R(\Fp)$ at the working prime, so $R(\Q)=\langle t\rangle$ is
cyclic of order $m$. Every rational class of degree $5$ lies on some $T_i=[5\infty]+it+A$, $0\le i<m$, and only the following
indices survive the coset test over $\Fp$: $i\in\{1,2,3,4\}$ for $N=79,89,101,131$; $\{1,2,3,4,23\}$ for $N=83$;
$\{2,3\}$ for $N=103,107,167,191$. For $g(Y)=1$ these are the families
$\pi^*\Pic^1(Y)+3P$, $\pi^*\Pic^2(Y)+P$, $\pi^*\Pic^2(Y)+Q$, $\pi^*\Pic^1(Y)+3Q$ with $P=\infty$,
$Q=w_N(\infty)$, and for $g(Y)=2$ only the middle two. The cusp is a fixed subdivisor: if $D$ contains no full
fibre of $\pi$ and $d+\deg D<g-2g(Y)+1$, then $H^0(C,\pi^*M(D))=H^0(Y,M)$ for every $M$ of degree $d$. At
$N=83$ the index $23$ is the family $\pi^*\Pic^1(Y)+D$ through an explicit $w_{83}$-fixed cubic point $D$: with
$F=\infty+w_{83}(\infty)$ and $c=[F+D-5\infty]$, invariance of $F+D$ under $w_{83}$, which acts as $-1$ on $R$,
gives $2c=5t$ in $R(\Q)\cong\Z/41$, so $c=23t$; the fixed subdivisor is $D$. At $79$ and $103$ the same sieve shows that all classes of $W^1_5(\Fp)$ lie over the indices $2,3$,
whose families have fixed cusps; so there is no function of degree $5$.

\subsection{Route T: known families (fifteen levels)}
For the remaining levels, $C$ has a double cover $C\to Y$ with $g(Y)=2$ or a map of degree $4$ to a curve of
genus $1$, giving a family $V=B+[F_4]+C\subseteq\W$ with $B$ the pulled-back Jacobian and $F_4$ a fibre (or
twice a fibre). For every tested subgroup $S$ and every full coset $x+S\subseteq\W(\Fp)$, the certificate checks (\cref{tab:routeT})
that $S\subseteq B(\Fp)$, that $x+S\subseteq B(\Fp)+[F_4]+C(\Fp)$, that $\dim T_xZ_S=\dim B+1$, and that the
differential of $B\times C\to\Pic^5(C)$ has rank $\dim B+1$. The subgroups tested cover every possible direction
in $\A$: for each prime order $\ell\ne p$ tested, every simple factor of $\A$ has Frobenius order divisible by
some tested $\ell$, and all subgroups of that order in $J(\Fp)$ are tested. At $82,99,128$ the unique positive
elliptic part is identified with $B$ through its newform, so the whole of $B(\Fp)$ may be used. At $86$ the
repeated elliptic factor is isogenous to $E_{43}\colon y^2+y=x^3+x^2$, which has no rational $2$-isogeny, so every
curve in its isogeny class is linked to it by an odd-degree isogeny and has cyclic group of order $8$ over $\F_7$;
all $192$ cyclic subgroups of order $8$ of $J(\F_7)$ are tested. By \cref{lem:local}, every translate $T$ lies in $V$.
Projecting to $J/B$, which has dimension $>1$, shows $T\subseteq B+[F_4]+P$ for a fixed point $P$, whose Galois
orbit is then a proper fixed subdivisor.

\begin{proof}[Proof of Theorem A]
Combine the function exclusions with the three routes, as in \cref{tab:A}. The $36$ certificates are in the
accompanying repository (\texttt{certificates/A\_x0\_quintic}), with a list of the data per level.
\end{proof}

\begin{table}[ht]\centering\small
\caption{Theorem A: the thirty levels. The route proves finiteness of the points of degree $5$ assuming there is
no function of degree $5$; the last column is the separate argument excluding functions of degree $5$ (``double
cover'': the inequality $g>2h+4$).}\label{tab:A}
\begin{tabular}{rrll@{\qquad}rrll}\toprule
$N$&$g$&route&degree-5 functions&$N$&$g$&route&degree-5 functions\\\midrule
74&8&T&local pencil, 2 primes&107&9&Q&double cover\\
77&7&T&local pencil&111&11&T&double cover\\
79&6&Q&quotient sieve&118&14&T&double cover\\
82&9&T&\cite{NajmanOrlic2024}&123&13&T&double cover\\
83&7&Q&double cover&128&9&T&$W^1_5(\F_3)=\varnothing$\\
85&7&R&local pencil&131&11&Q&double cover\\
86&10&T&$W^1_5(\F_7)=\varnothing$&141&15&T&double cover\\
88&9&R&local pencil&142&17&T&double cover\\
89&7&Q&double cover&143&13&T&double cover\\
92&10&R&double cover&145&13&T&double cover\\
96&9&R&rank zero&155&15&T&double cover\\
98&7&R&rank zero&159&17&T&double cover\\
99&9&T&$W^1_5(\F_5)=\varnothing$&167&14&Q&double cover\\
100&7&R&rank zero&191&16&Q&double cover\\
101&8&Q&double cover&&&&\\
103&8&Q&quotient sieve&&&&\\\bottomrule
\end{tabular}
\end{table}
\section{Points of degree 5 on intermediate curves: proof of Theorem B}\label{sec:quintic-intermediate}

\subsection{Reduction to finitely many curves}
\begin{lemma}[Projection]\label{lem:projection}
Let $C\to D$ be a finite morphism over $\Q$. If $D$ has only finitely many points of degree $1$ and of degree
$5$, then $C$ has only finitely many points of degree $5$.
\end{lemma}
\begin{proof}
The image of a point of degree $5$ has degree dividing $5$, and fibres are finite.
\end{proof}
Forgetful maps $X_\Delta(N)\to X_{\Delta'}(M)$, for $M\mid N$ and $\Delta$ mapping into $\Delta'$, are defined
over $\Q$; in particular every $X_\Delta(N)$ maps to $X_0(N)$. By Theorem~A and \cite[Theorem~1.10]{DHJO},
$X_0(N)$ has finitely many points of degree $5$ unless
\[
 N\in\{1,\dots,45\}\cup\{47,\dots,58\}\cup\{61,63,64,65,67,68,72,73,75,80,81,91,109,121,125\},
\]
and every $X_0(N)$ with $N$ outside this list has genus $>1$ (otherwise its rational cusp would give a
function of degree $5$, \S\ref{sec:method-I}), hence finitely many rational points by Faltings
\cite{Faltings1991}; so these $X_0(N)$ can serve as targets in \cref{lem:projection}. So every strict pair with $N\ge192$, and $807$ of the $1011$
strict pairs with $N\le191$, have finitely many points of degree $5$. The strict pairs with $N\le191$ are
enumerated twice, once from primitive rows $(c,d)$ modulo $N$ up to $\Delta$ and once with Sage's
\texttt{GammaH}; full sets of elements, not orders or genera, are used to identify them. Seven more pairs map to
intermediate curves already shown to be negative (\cref{tab:projection}), and fifteen more, of genus $57$ to
$256$, satisfy $119\mu>120000$, so they have $\gon_{\mathbb C}>10$ and finitely many points of degree $5$ by
\cref{lem:frey} and \cite[Proposition~2.2]{DerickxSutherland2017}. This leaves $182$ strict pairs.

\subsection{The \texorpdfstring{$43$}{43} positive curves}
Each curve in \cref{tab:positive} has a function of degree $5$ over $\Q$, hence infinitely many points of degree
$5$ (\cref{lem:hilbert}):
\begin{itemize}
\item[L] ($13$ curves, $g\le2$): for a rational cusp $P$, $\ell(5P)=6-g>\ell(4P)$;
\item[K] ($14$ curves, $g=3,4$): in genus $3$, $5$ is not a Weierstrass gap at the cusp $\infty$ (read off
 from the orders of vanishing of a basis of cusp forms); in genus $4$ the curve is not hyperelliptic (the
 quadrics in the canonical ideal have the right rank), so $K-P$ is base-point free of degree $5$;
\item[W] ($12$ curves): an explicit divisor $D$ of degree $5$ with $\ell(D)=2$ and no base point, on a canonical
 model certified through the Sturm bound and a Hilbert polynomial comparison;
\item[P] ($4$ curves): the $\Q$-pentagonal curves of \cite[Theorems~1.4--1.5]{Orlic2025}.
\end{itemize}

\subsection{The negative curves}
For the remaining $139$ curves we show finiteness. The arguments are listed by code in \cref{tab:negative}.

\begin{lemma}[Rank zero]\label{lem:rankzero}
If $C(\Q)\ne\varnothing$, $J(C)(\Q)$ is finite and $C$ has no function of degree $5$ over $\Q$, then $C$ has
finitely many points of degree $5$.
\end{lemma}
\begin{proof}
As in \cref{lem:inject}, each point of degree $5$ is the unique effective divisor in its class, and there are
finitely many classes.
\end{proof}

\emph{Rank zero} ($82$ curves). We show $J(\Q)$ finite from nonvanishing $L$-values (exact modular-symbol
$L$-ratios), using Kato \cite[Corollary~14.3]{Kato2004} as applied to modular Jacobians with nontrivial
character in \cite[\S4]{DerickxSutherland2017}. The absence of a function of degree $5$ comes from
\begin{itemize}
\item[Z] ($74$ curves) Orli\'c's classification: genus $\ge5$ and not in Orli\'c's lists of curves of
 $\Q$-gonality at most $5$, so $\gon_\Q\ge6$ \cite{Orlic2025};
\item[B] ($2$ curves, $35\langle-1,8\rangle$ and $55\langle-1,4\rangle$, of genus $7$ and $9$) the curves are
 bielliptic \cite[Theorem~0.2]{JeonKimSchweizer2020}, so a function of degree $5$ would violate
 \cref{lem:CS}: $g\le 2+4=6$;
\item[T] ($3$ curves) Petri lifting: over $\Fp$ every class of $W^1_5$ is $A+P$ for an explicit rational
 base-point free class $A$ of degree $4$, with Petri tangent dimension $1$; \cref{lem:local} identifies $W^1_5$
 near each such class with the Abel curve $A+C$, so a rational pencil of degree $5$ has a base point;
\item[R] ($3$ curves) a complete sieve of rational divisor classes: an explicit subgroup of $J(\Q)$ generated
 by differences of rational points is shown to be all of $J(\Q)$, and every class of $W^1_5(\Fp)$ in its reduction
 is lifted and checked over $\Q$. At level $40$ the explicit subgroup has order $4032$, and $\#J(\Q)$ is bounded
 above by the Hecke method of \cite[\S4.3]{DEH2021}: rational torsion is fixed by complex conjugation, and for
 $\ell\nmid40$ the Eichler--Shimura relation shows that $1-T_\ell+\ell\langle\ell\rangle$ kills rational torsion of order
 prime to $\ell$; both operators act on the integral homology, computed from Manin symbols. Since the relation
 at $\ell$ says nothing about $\ell$-primary torsion, for each $\ell\in\{3,7,11,13\}$ we bound the $\ell$-primary
 part using only the relations at the other three primes, and the remaining primary parts using all four; each
 of these computations gives a group of order $8064=2^7\cdot3^2\cdot7$. Point counts over finite fields give
 $\#J(\Q)\mid28224$, so $\#J(\Q)\mid\gcd(8064,28224)=4032$; at level $48$ the bounds $2304$ and $12800$ come from the
 description of $J_1(48)(\Q)_{\mathrm{tors}}$ by cuspidal divisors \cite[Theorem~4.13]{DEH2021}, transferred along
 the forgetful map.
\end{itemize}

\emph{Positive rank} ($57$ curves). None of these curves has a function of degree $5$ over $\Q$: they have genus
at least $8$ and are not in the classification of curves of $\Q$-gonality at most $5$
\cite{IshiiMomose1991,JeonKim2007}, \cite[Theorems~1.3--1.5]{Orlic2025}, except $43\langle-1,2\rangle$, which has
$\Q$-gonality $7$ by \cref{prop:43}. Here we use the classification of curves with infinitely many points of
degree at most $4$ \cite[Theorems~1.4--1.6]{DerickxOrlic2026}: none of these curves has infinitely many points of
degree $\le4$, so if it had infinitely many points of degree $5$, its minimum density degree $\min\delta(X/\Q)$ would be $5$. By Kadets and
Vogt \cite[Theorem~1.3]{KadetsVogt2025}, applied with $d=5$ (so $m=\lceil d/2\rceil-1=2$ and $\varepsilon=3d-1-6m=2$),
either $g\le\max(d(d-1)/2+1,\,3m(m-1)+m\varepsilon)=11$, or there is a morphism $X\to Y$ of degree $5$ to a curve
$Y$ with infinitely many rational points, that is, to $\PP^1$ or to an elliptic curve of positive rank. So for
$g\ge12$ it suffices to exclude these two kinds of maps. For $9\le g\le11$, the classification of curves of minimum density degree $5$
\cite[Proposition~6.3 and Table~1]{KadetsVogt2025} leaves, besides these covers (and curves of genus at most $8$),
only normalizations of Debarre--Fahlaoui curves: the points may then come from a curve in
$\mathrm{Sym}^2E$ of class $(5+j)H-jF$ with $1\le j\le5$ (Debarre--Fahlaoui \cite{DebarreFahlaoui1993}, \cite[Definition~5.1]{KadetsVogt2025}); adjunction gives
arithmetic genus $11-j(j-1)/2$, so $j\le2$ and the addition map to $E$ has degree $6$ or $7$.
\begin{itemize}
\item[E] ($43$ curves, genus $\ge12$, so only elliptic maps of degree $5$ need to be excluded) every positive-rank elliptic factor $E$ has conductor $N$, multiplicity one
 and odd analytic rank, and for $N\le800$ the parametrization through $X_0(N)$ is optimal
 \cite[Corollary~4.10]{DerickxOrlic2026}; so every morphism $X_\Delta(N)\to E'$ with $E'$ isogenous to $E$ has
 degree divisible by $t\cdot d_E$, where $t=[(\Z/N\Z)^\times:\Delta]$ and $d_E$ is the optimal modular degree,
 and $t,d_E>1$ in all these cases (\cref{tab:classE}; see also \cite[Corollary~4.12]{DerickxOrlic2026}); so $t\,d_E$ is composite and
 the degree is not $5$;
\item[E0] ($5$ curves, at levels $109$ and $125$) the decomposition of the Jacobian has no positive-rank
 elliptic factor;
\item[D] ($3$ curves of genus $9$ or $10$: $37\langle-1,10\rangle$, $57\langle-1,2\rangle$, $73\langle-1,6\rangle$)
 here a Debarre--Fahlaoui configuration must also be excluded, that is, elliptic maps of degree $5$, $6$ or $7$:
 at level $73$ there is no positive-rank elliptic factor, and at levels $37$ and $57$ every elliptic map of
 positive rank has degree at least $12$ and $8$ respectively \cite[Corollary~4.10]{DerickxOrlic2026};
\item[C] ($1$ curve, $43\langle-1,2\rangle$) the only possibly positive-rank factor is an elliptic curve $A$; at
 $p=2$, $A(\F_2)$ has order $5$, and no full coset of it lies in $\W(\F_2)$, which has $2133$ elements. A
 positive-dimensional translate in $\W$ over $\Q$ would reduce to such a coset (prime-to-$2$ torsion specializes
 injectively), so $\W(\Q)$ is finite;
\item[F] ($5$ curves, levels $53$, $61$, $65$) there is a map $f\colon C\to E$ of degree $4$ whose pullback is
 the positive-rank part $A$, and the surface $Z=\{f^*Q+P\}$ of reducible divisors lies in $\W$. At a good odd
 prime $p$, with $B=A(\Fp)$, we check that every full $B$-coset in $\W(\Fp)$ lies in $Z$, that the intersection
 of the corresponding tangent spaces has dimension at most $2$, and that $A\times C\to\Pic^5C$ has injective
 differential at the relevant points. Since $f_*f^*=[4]$ and $p$ is odd, $A$ has a smooth model over $\Z_p$;
 lifting $B$ to $A(\Z_p)$ and applying \cref{lem:local} puts every rational translate of $A$ in $\W$ into $Z$.
 A point of degree $5$ in $Z$ would be $f^*Q+P$ with a unique, hence rational, point $P$, which is impossible.
\end{itemize}

\begin{proof}[Proof of Theorem B]
The $43$ positive curves, the $139$ curves above, the $7$ further projections, the $15$ curves with large index
and the $807$ projections to $X_0(N)$ account for all $1011$ strict pairs with $N\le191$, and projection to
$X_0(N)$ handles $N\ge192$.
\end{proof}

\begin{proof}[Proof of Corollary E]
The $39$ positive curves other than Orli\'c's four $\Q$-pentagonal curves have $\Q$-gonality at most $4$: those
of genus $g\le4$ have a rational cusp, so $\gon_\Q\le\max(g,2)\le4$ (\S\ref{sec:method-I}), and the twelve
curves of type W are $\Q$-tetragonal \cite[Theorem~1.3]{Orlic2025}. By \cref{lem:hilbert} they have density degree at most $4$. The four
pentagonal curves have genus $6$, $6$, $8$ and $16$, so finitely many rational points by Faltings
\cite{Faltings1991}, and none of them occurs in the lists of \cite[Theorems~1.4, 1.5 and~1.6]{DerickxOrlic2026}
of curves with infinitely many points of degree $2$, $3$ or $4$. By Theorem~B they have infinitely many points
of degree $5$.
\end{proof}

Every case has a certificate or a precise citation in the accompanying repository
(\texttt{certificates/\allowbreak B\_intermediate\_quintic}), which also contains the full case list.
\section{Computations}\label{sec:computations}

All computations use Magma~V2.29-4 \cite{Magma}, SageMath~10.7 with MD~Sage \cite{MDSage} (commit
\texttt{6022429}), CurveArith \cite{DerickxTerao} (commit \texttt{b7c6192}) and Zywina's package
\cite{ZywinaModular} (commit \texttt{ce31d78}); the three packages are pinned as git submodules of the
accompanying repository. The repository is organized by theorem:
\begin{center}\small
\begin{tabular}{lp{0.45\textwidth}}\toprule
folder & contents\\\midrule
\texttt{data/} & one row per curve: bounds, methods, certificates or citations\\
\texttt{certificates/A\_x0\_quintic/} & Theorem A: $36$ Magma certificates and logs\\
\texttt{certificates/B\_intermediate\_quintic/} & Theorem B and Corollary E: all $1388$ pairs $(N,\Delta)$, $N\le191$ ($1011$ strict)\\
\texttt{certificates/C\_intermediate\_gonality/} & Theorem C: models, CurveArith runs, upper-bound certificates\\
\texttt{certificates/C\_43.132.9.a.2/} & \cref{prop:43}\\
\texttt{certificates/D\_x1\_gonality/} & Theorem D: Phase~1 driver, Phase~2 program, logs\\
\texttt{verify.py} & checks that every row of \texttt{data/} has a passing log or a citation\\\bottomrule
\end{tabular}
\end{center}

\subsection*{Method II for \texorpdfstring{$X_1(N)$}{X1(N)}}
Phase~1 is a short Magma script that loads CurveArith and calls its internal function
\texttt{PrecomputePowerseriesExpansions} with exactly the places and precisions that
\texttt{CAHasFunctionOfDegreeAtMost} would use; no CurveArith code is modified. It runs in several Magma processes. Every
process enumerates all places, as CurveArith does, and computes the expansions only at the places assigned to it: the
place's key (the irreducible polynomial $g(t)\in\Fp[t]$ below it, encoded as an integer, or $0$ at infinity) is
hashed by a fixed integer function, and the hash determines the process. This assignment does not depend on the order in which Magma lists the places, so
the processes together treat every place exactly once; the merge checks that the counts add up, and that the
processes agree on the curve, the place counts and the basis of differentials. (Splitting by position in the list
instead is not safe: Magma does not always list the places in the same order in different processes.) Its
output is the list of matrices of \cref{lem:rank}.

Phase~2 is a C++ program of about $1000$ lines (with GMP) that enumerates $\mathcal F_b(d)$, performs the
incremental Gaussian elimination of \S\ref{sec:phase2} over $\Fp$, and enumerates the kernel vectors up to scalars
(\cref{lem:kernel}). It is a reimplementation, for speed, of the option \texttt{Al:="Cover"} of a modified
CurveArith, which we used to cross-check it on small curves. The baseline $b$ is chosen to minimize the number of
divisors times the size of the kernels. For any $b$ the result is correct; it only affects the running time.

\subsection*{Validation}
The pipeline was tested in both directions against CurveArith's unmodified \texttt{CAGonality}: on
$X_1(26)$ over $\F_3,\F_5,\F_7$ and on $X_1(29)$ over $\F_2,\F_3$, it finds no function of degree
$\gon_{\F_p}-1$ and finds one of degree $\gon_{\F_p}$. When a function is found, its divisor is checked independently by a rank computation.
Each Phase~1 process also computes a few Riemann--Roch dimensions with Magma's \texttt{Dimension}, for
divisors supported on its own places; Phase~2 checks them against \cref{lem:rank} before searching.

\subsection*{Running times}
The $36$ certificates of Theorem~A take $40$ minutes of CPU time in total, the longest ($N=86$) $16$ minutes.
The certificates of \cref{prop:43} take a few minutes. The lower bounds of Theorem~C take about $2$ hours in
total, dominated by $48.384.19.bc.1$ ($66$ minutes). The running times for Theorem~D are in \cref{tab:D}.

\subsection*{Acknowledgements}
The computations rest on software written by others: I thank Maarten Derickx for MD~Sage and for permission to build on its code,
Maarten Derickx and Kenji Terao for CurveArith, Andrew Sutherland and Mark van Hoeij for their optimized
models of $X_1(N)$, David Zywina for his Magma package for modular curves, and the LMFDB collaboration,
whose database was used to find examples and to name curves; its model of one curve is used in
\cref{prop:43}, and otherwise the LMFDB is not a source of proofs. I thank Petar
Orli\'c for the tables of \cite{Orlic2025}, on which much of this paper builds.

\subsection*{Use of AI tools}
Large language model assistants (OpenAI's GPT-6 Astra, through Codex, and Anthropic's Claude, through Claude Code)
were used to write and optimize code, to organize the certificates, and to draft parts of the text. The proofs
are built to be checked without trusting these tools: each step is a cited published result, a short
argument written out in full, or a computation backed by a certificate in the accompanying repository that can be rerun independently. The author
takes responsibility for the content of this paper.

\bibliographystyle{amsplain}
\bibliography{refs}

\appendix
\section{Tables}\label{sec:tables}

\begin{longtable}{llrrll}
\caption{$\Q$-gonality of the strict curves $X_\Delta(N)$, $N\le50$. Lower and upper bound: method
 (\S\ref{sec:methods}), or ``genus'' for $g\le1$ or the low-genus arguments of \S\ref{sec:method-I}.
 For $N\le40$ and $g\ge3$ both bounds are from \cite[Theorem~1.2, Table~1]{Orlic2025}.}\label{tab:C}\\
\toprule label & $\Delta$ & $g$ & $\gon_\Q$ & lower bound & upper bound\\\midrule\endfirsthead
\toprule label & $\Delta$ & $g$ & $\gon_\Q$ & lower bound & upper bound\\\midrule\endhead
13.28.0.a.1 & $\langle -1,3\rangle$ & 0 & 1 & genus & genus\\
13.42.0.a.2 & $\langle -1,5\rangle$ & 0 & 1 & genus & genus\\
15.48.1.a.1 & $\langle -1,4\rangle$ & 1 & 2 & genus & genus\\
16.48.0.v.1 & $\langle -1,7\rangle$ & 0 & 1 & genus & genus\\
17.36.1.a.2 & $\langle -1,2\rangle$ & 1 & 2 & genus & genus\\
17.72.1.a.2 & $\langle -1,4\rangle$ & 1 & 2 & genus & genus\\
19.60.1.a.2 & $\langle -1,7\rangle$ & 1 & 2 & genus & genus\\
20.72.1.f.2 & $\langle -1,9\rangle$ & 1 & 2 & genus & genus\\
21.64.1.a.3 & $\langle -1,4\rangle$ & 1 & 2 & genus & genus\\
21.96.3.a.1 & $\langle -1,8\rangle$ & 3 & 2 & \cite{Orlic2025} & \cite{Orlic2025}\\
24.96.1.dg.2 & $\langle -1,11\rangle$ & 1 & 2 & genus & genus\\
24.96.3.ge.2 & $\langle -1,5\rangle$ & 3 & 3 & \cite{Orlic2025} & \cite{Orlic2025}\\
24.96.3.gi.1 & $\langle -1,7\rangle$ & 3 & 3 & \cite{Orlic2025} & \cite{Orlic2025}\\
25.60.0.a.1 & $\langle -1,4\rangle$ & 0 & 1 & genus & genus\\
25.150.4.f.1 & $\langle -1,7\rangle$ & 4 & 4 & \cite{Orlic2025} & \cite{Orlic2025}\\
26.84.4.a.1 & $\langle -1,3\rangle$ & 4 & 3 & \cite{Orlic2025} & \cite{Orlic2025}\\
26.126.4.a.1 & $\langle -1,5\rangle$ & 4 & 3 & \cite{Orlic2025} & \cite{Orlic2025}\\
27.108.1.a.1 & $\langle -1,8\rangle$ & 1 & 2 & genus & genus\\
28.96.4.c.4 & $\langle -1,3\rangle$ & 4 & 3 & \cite{Orlic2025} & \cite{Orlic2025}\\
28.144.4.d.1 & $\langle -1,13\rangle$ & 4 & 3 & \cite{Orlic2025} & \cite{Orlic2025}\\
29.60.4.a.1 & $\langle -1,4\rangle$ & 4 & 3 & \cite{Orlic2025} & \cite{Orlic2025}\\
29.210.8.b.2 & $\langle -1,12\rangle$ & 8 & 6 & \cite{Orlic2025} & \cite{Orlic2025}\\
30.144.5.a.1 & $\langle -1,11\rangle$ & 5 & 4 & \cite{Orlic2025} & \cite{Orlic2025}\\
31.96.6.a.1 & $\langle -1,2\rangle$ & 6 & 5 & \cite{Orlic2025} & \cite{Orlic2025}\\
31.160.6.c.1 & $\langle -1,5\rangle$ & 6 & 5 & \cite{Orlic2025} & \cite{Orlic2025}\\
32.96.1.f.1 & $\langle -1,7\rangle$ & 1 & 2 & genus & genus\\
32.192.5.bu.1 & $\langle -1,15\rangle$ & 5 & 4 & \cite{Orlic2025} & \cite{Orlic2025}\\
33.96.5.a.4 & $\langle -1,2\rangle$ & 5 & 4 & \cite{Orlic2025} & \cite{Orlic2025}\\
33.240.11.a.1 & $\langle -1,10\rangle$ & 11 & 6 & \cite{Orlic2025} & \cite{Orlic2025}\\
34.108.5.a.2 & $\langle -1,9\rangle$ & 5 & 4 & \cite{Orlic2025} & \cite{Orlic2025}\\
34.216.9.a.1 & $\langle -1,13\rangle$ & 9 & 6 & \cite{Orlic2025} & \cite{Orlic2025}\\
35.96.5.a.2 & $\langle -1,4\rangle$ & 5 & 4 & \cite{Orlic2025} & \cite{Orlic2025}\\
35.144.7.a.2 & $\langle -1,8\rangle$ & 7 & 4 & \cite{Orlic2025} & \cite{Orlic2025}\\
35.192.9.b.3 & $\langle -1,11\rangle$ & 9 & 6 & \cite{Orlic2025} & \cite{Orlic2025}\\
35.288.13.a.2 & $\langle -1,6\rangle$ & 13 & 8 & \cite{Orlic2025} & \cite{Orlic2025}\\
36.144.3.c.1 & $\langle -1,11\rangle$ & 3 & 3 & \cite{Orlic2025} & \cite{Orlic2025}\\
36.216.7.u.1 & $\langle -1,17\rangle$ & 7 & 4 & \cite{Orlic2025} & \cite{Orlic2025}\\
37.76.4.a.2 & $\langle -1,3\rangle$ & 4 & 3 & \cite{Orlic2025} & \cite{Orlic2025}\\
37.114.4.b.1 & $\langle -1,8\rangle$ & 4 & 3 & \cite{Orlic2025} & \cite{Orlic2025}\\
37.228.10.a.2 & $\langle -1,10\rangle$ & 10 & 6 & \cite{Orlic2025} & \cite{Orlic2025}\\
37.342.16.c.2 & $\langle -1,6\rangle$ & 16 & 9 & \cite{Orlic2025} & \cite{Orlic2025}\\
38.180.10.a.1 & $\langle -1,7\rangle$ & 10 & 6 & \cite{Orlic2025} & \cite{Orlic2025}\\
39.112.5.a.2 & $\langle -1,4\rangle$ & 5 & 4 & \cite{Orlic2025} & \cite{Orlic2025}\\
39.168.9.a.1 & $\langle -1,5\rangle$ & 9 & 4 & \cite{Orlic2025} & \cite{Orlic2025}\\
39.224.9.a.3 & $\langle -1,16\rangle$ & 9 & 6 & \cite{Orlic2025} & \cite{Orlic2025}\\
39.336.17.c.1 & $\langle -1,14\rangle$ & 17 & 8 & \cite{Orlic2025} & \cite{Orlic2025}\\
40.144.5.gn.1 & $\langle -1,9,11\rangle$ & 5 & 4 & \cite{Orlic2025} & \cite{Orlic2025}\\
40.144.7.fp.1 & $\langle -1,3\rangle$ & 7 & 4 & \cite{Orlic2025} & \cite{Orlic2025}\\
40.144.7.fs.1 & $\langle -1,7\rangle$ & 7 & 4 & \cite{Orlic2025} & \cite{Orlic2025}\\
40.288.13.sp.1 & $\langle -1,9\rangle$ & 13 & 8 & \cite{Orlic2025} & \cite{Orlic2025}\\
40.288.13.sq.2 & $\langle -1,11\rangle$ & 13 & 7 & \cite{Orlic2025} & \cite{Orlic2025}\\
40.288.9.bh.1 & $\langle -1,19\rangle$ & 9 & 6 & \cite{Orlic2025} & \cite{Orlic2025}\\
41.84.5.a.1 & $\langle -1,2\rangle$ & 5 & 4 & II, $p=3$ & I (pullback)\\
41.168.11.b.1 & $\langle -1,4\rangle$ & 11 & 6 & II, $p=2$ & I (cusps)\\
41.210.11.b.1 & $\langle -1,3\rangle$ & 11 & 8 & II, $p=5$ & I (cusps)\\
41.420.21.d.1 & $\langle -1,9\rangle$ & 21 & 11 & II, $p=2$ & I (cusps)\\
42.192.9.c.2 & $\langle -1,5\rangle$ & 9 & 6 & II, $p=5$ & I (pullback)\\
42.288.13.a.1 & $\langle -1,13\rangle$ & 13 & 6 & II, $p=5$ & I (pullback)\\
43.132.9.a.2 & $\langle -1,2\rangle$ & 9 & 7 & II+III, $p=3$ & I (special)\\
43.308.15.b.1 & $\langle -1,6\rangle$ & 15 & 9 & II, $p=2$ & I (special)\\
44.144.8.c.4 & $\langle -1,5\rangle$ & 8 & 5 & II, $p=3$ & \cite{Orlic2025}\\
44.360.16.e.1 & $\langle -1,21\rangle$ & 16 & 8 & II, $p=3$ & I (pullback)\\
45.144.5.a.1 & $\langle -1,4\rangle$ & 5 & 4 & II, $p=2$ & \cite{Orlic2025}\\
45.216.11.d.2 & $\langle -1,8\rangle$ & 11 & 6 & II, $p=2$ & I (pullback)\\
45.288.9.a.4 & $\langle -1,14\rangle$ & 9 & 6 & II, $p=2$ & I (pullback)\\
45.432.21.g.2 & $\langle -1,19\rangle$ & 21 & 10 & II, $p=2$ & I (cusps)\\
48.192.5.ky.2 & $\langle -1,11\rangle$ & 5 & 4 & II, $p=5$ & I (pullback)\\
48.192.7.hj.1 & $\langle -1,5\rangle$ & 7 & 4 & II, $p=5$ & I (pullback)\\
48.192.7.ho.1 & $\langle -1,7,17\rangle$ & 7 & 4 & II, $p=5$ & I (pullback)\\
48.384.13.nt.1 & $\langle -1,23\rangle$ & 13 & 8 & II, $p=7$ & I (pullback)\\
48.384.19.bc.1 & $\langle -1,7\rangle$ & 19 & 8 & II, $p=7$ & I (pullback)\\
48.384.19.bj.1 & $\langle -1,17\rangle$ & 19 & 8 & II, $p=5$ & I (pullback)\\
49.168.3.b.1 & $\langle -1,6\rangle$ & 3 & 3 & genus & genus\\
49.392.19.i.2 & $\langle -1,18\rangle$ & 19 & 11 & II, $p=2$ & I (cusps)\\
50.180.4.a.2 & $\langle -1,9\rangle$ & 4 & 3 & genus & I (pullback)\\
50.450.22.f.1 & $\langle -1,7\rangle$ & 22 & 10 & II, $p=3$ & I (pullback)\\
\bottomrule
\end{longtable}

\begingroup\small
\begin{longtable}{rrrl}
\caption{Positive strict intermediate pairs.}\label{tab:positive}\\
\toprule
$N$ & Generators of $\Delta$ & $g$ & Proof \\
\midrule\endfirsthead
\toprule $N$ & Generators of $\Delta$ & $g$ & Proof \\
\midrule\endhead
13 & $\langle -1, 3\rangle$ & 0 & L \\
13 & $\langle -1, 5\rangle$ & 0 & L \\
15 & $\langle -1, 4\rangle$ & 1 & L \\
16 & $\langle -1, 7\rangle$ & 0 & L \\
17 & $\langle -1, 2\rangle$ & 1 & L \\
17 & $\langle -1, 4\rangle$ & 1 & L \\
19 & $\langle -1, 7\rangle$ & 1 & L \\
20 & $\langle -1, 9\rangle$ & 1 & L \\
21 & $\langle -1, 4\rangle$ & 1 & L \\
21 & $\langle -1, 8\rangle$ & 3 & K \\
24 & $\langle -1, 5\rangle$ & 3 & K \\
24 & $\langle -1, 7\rangle$ & 3 & K \\
24 & $\langle -1, 11\rangle$ & 1 & L \\
25 & $\langle -1, 4\rangle$ & 0 & L \\
25 & $\langle -1, 7\rangle$ & 4 & K \\
26 & $\langle -1, 3\rangle$ & 4 & K \\
26 & $\langle -1, 5\rangle$ & 4 & K \\
27 & $\langle -1, 8\rangle$ & 1 & L \\
28 & $\langle -1, 3\rangle$ & 4 & K \\
28 & $\langle -1, 13\rangle$ & 4 & K \\
29 & $\langle -1, 4\rangle$ & 4 & K \\
30 & $\langle -1, 11\rangle$ & 5 & W \\
31 & $\langle -1, 2\rangle$ & 6 & P \\
31 & $\langle -1, 5\rangle$ & 6 & P \\
32 & $\langle -1, 7\rangle$ & 1 & L \\
32 & $\langle -1, 15\rangle$ & 5 & W \\
33 & $\langle -1, 2\rangle$ & 5 & W \\
34 & $\langle -1, 9\rangle$ & 5 & W \\
35 & $\langle -1, 4\rangle$ & 5 & W \\
36 & $\langle -1, 11\rangle$ & 3 & K \\
37 & $\langle -1, 3\rangle$ & 4 & K \\
37 & $\langle -1, 8\rangle$ & 4 & K \\
39 & $\langle -1, 4\rangle$ & 5 & W \\
40 & $\langle -1, 7\rangle$ & 7 & W \\
40 & $\langle -1, 9, 11\rangle$ & 5 & W \\
41 & $\langle -1, 2\rangle$ & 5 & W \\
44 & $\langle -1, 5\rangle$ & 8 & P \\
45 & $\langle -1, 4\rangle$ & 5 & W \\
48 & $\langle -1, 11\rangle$ & 5 & W \\
49 & $\langle -1, 6\rangle$ & 3 & K \\
50 & $\langle -1, 9\rangle$ & 4 & K \\
64 & $\langle -1, 7\rangle$ & 5 & W \\
125 & $\langle -1, 4\rangle$ & 16 & P \\
\bottomrule
\end{longtable}
\endgroup

\begingroup\small
\begin{longtable}{rrrrl}
\caption{All direct negative assignments (excluding projection).}\label{tab:negative}\\
\toprule $N$ & Generators of $\Delta$ & $g$ & $\mu$ & Proof \\
\midrule\endfirsthead
\toprule $N$ & Generators of $\Delta$ & $g$ & $\mu$ & Proof \\
\midrule\endhead
29 & $\langle -1, 12\rangle$ & 8 & 210 & Z \\
33 & $\langle -1, 10\rangle$ & 11 & 240 & Z \\
34 & $\langle -1, 13\rangle$ & 9 & 216 & Z \\
35 & $\langle -1, 8\rangle$ & 7 & 144 & B \\
35 & $\langle -1, 11\rangle$ & 9 & 192 & Z \\
35 & $\langle -1, 6\rangle$ & 13 & 288 & Z \\
36 & $\langle -1, 17\rangle$ & 7 & 216 & T \\
37 & $\langle -1, 10\rangle$ & 10 & 228 & D \\
37 & $\langle -1, 6\rangle$ & 16 & 342 & E \\
38 & $\langle -1, 7\rangle$ & 10 & 180 & Z \\
39 & $\langle -1, 5\rangle$ & 9 & 168 & T \\
39 & $\langle -1, 16\rangle$ & 9 & 224 & Z \\
39 & $\langle -1, 14\rangle$ & 17 & 336 & Z \\
40 & $\langle -1, 3\rangle$ & 7 & 144 & R \\
40 & $\langle -1, 9\rangle$ & 13 & 288 & Z \\
40 & $\langle -1, 11\rangle$ & 13 & 288 & Z \\
40 & $\langle -1, 19\rangle$ & 9 & 288 & Z \\
41 & $\langle -1, 4\rangle$ & 11 & 168 & Z \\
41 & $\langle -1, 3\rangle$ & 11 & 210 & Z \\
41 & $\langle -1, 9\rangle$ & 21 & 420 & Z \\
42 & $\langle -1, 5\rangle$ & 9 & 192 & Z \\
42 & $\langle -1, 13\rangle$ & 13 & 288 & Z \\
43 & $\langle -1, 2\rangle$ & 9 & 132 & C \\
43 & $\langle -1, 6\rangle$ & 15 & 308 & E \\
44 & $\langle -1, 21\rangle$ & 16 & 360 & Z \\
45 & $\langle -1, 8\rangle$ & 11 & 216 & Z \\
45 & $\langle -1, 14\rangle$ & 9 & 288 & Z \\
45 & $\langle -1, 19\rangle$ & 21 & 432 & Z \\
48 & $\langle -1, 5\rangle$ & 7 & 192 & R \\
48 & $\langle -1, 7, 17\rangle$ & 7 & 192 & R \\
48 & $\langle -1, 7\rangle$ & 19 & 384 & Z \\
48 & $\langle -1, 17\rangle$ & 19 & 384 & Z \\
48 & $\langle -1, 23\rangle$ & 13 & 384 & Z \\
49 & $\langle -1, 18\rangle$ & 19 & 392 & Z \\
50 & $\langle -1, 7\rangle$ & 22 & 450 & Z \\
51 & $\langle -1, 2\rangle$ & 9 & 144 & Z \\
51 & $\langle -1, 4\rangle$ & 17 & 288 & Z \\
51 & $\langle -1, 16\rangle$ & 33 & 576 & Z \\
52 & $\langle -1, 3\rangle$ & 9 & 168 & Z \\
52 & $\langle -1, 5\rangle$ & 13 & 252 & Z \\
52 & $\langle -1, 9\rangle$ & 19 & 336 & Z \\
52 & $\langle -1, 25\rangle$ & 25 & 504 & Z \\
53 & $\langle -1, 4\rangle$ & 8 & 108 & F \\
53 & $\langle -1, 23\rangle$ & 40 & 702 & E \\
54 & $\langle -1, 17\rangle$ & 10 & 324 & Z \\
55 & $\langle -1, 4\rangle$ & 9 & 144 & B \\
55 & $\langle -1, 16\rangle$ & 17 & 288 & Z \\
55 & $\langle -1, 12\rangle$ & 21 & 360 & Z \\
55 & $\langle -1, 21\rangle$ & 41 & 720 & Z \\
56 & $\langle -1, 3\rangle$ & 9 & 192 & Z \\
56 & $\langle -1, 5\rangle$ & 11 & 192 & Z \\
56 & $\langle -1, 17\rangle$ & 11 & 192 & Z \\
56 & $\langle -1, 13, 15\rangle$ & 13 & 288 & Z \\
56 & $\langle -1, 9\rangle$ & 21 & 384 & Z \\
56 & $\langle -1, 13\rangle$ & 31 & 576 & Z \\
56 & $\langle -1, 15\rangle$ & 31 & 576 & Z \\
56 & $\langle -1, 27\rangle$ & 25 & 576 & Z \\
57 & $\langle -1, 2\rangle$ & 9 & 160 & D \\
57 & $\langle -1, 11\rangle$ & 13 & 240 & E \\
57 & $\langle -1, 7\rangle$ & 25 & 480 & E \\
57 & $\langle -1, 20\rangle$ & 43 & 720 & E \\
58 & $\langle -1, 5\rangle$ & 12 & 180 & E \\
58 & $\langle -1, 17\rangle$ & 36 & 630 & E \\
61 & $\langle -1, 4\rangle$ & 8 & 124 & F \\
61 & $\langle -1, 8\rangle$ & 12 & 186 & E \\
61 & $\langle -1, 21\rangle$ & 16 & 310 & E \\
61 & $\langle -1, 3\rangle$ & 26 & 372 & E \\
61 & $\langle -1, 13\rangle$ & 36 & 620 & E \\
61 & $\langle -1, 11\rangle$ & 56 & 930 & E \\
63 & $\langle -1, 4, 5\rangle$ & 9 & 192 & Z \\
63 & $\langle -1, 2\rangle$ & 17 & 288 & Z \\
63 & $\langle -1, 11\rangle$ & 17 & 288 & Z \\
63 & $\langle -1, 10\rangle$ & 17 & 288 & Z \\
63 & $\langle -1, 13\rangle$ & 13 & 288 & Z \\
63 & $\langle -1, 4\rangle$ & 33 & 576 & Z \\
63 & $\langle -1, 17\rangle$ & 33 & 576 & Z \\
63 & $\langle -1, 20\rangle$ & 25 & 576 & Z \\
63 & $\langle -1, 8\rangle$ & 49 & 864 & Z \\
64 & $\langle -1, 15\rangle$ & 13 & 384 & Z \\
64 & $\langle -1, 31\rangle$ & 37 & 768 & Z \\
65 & $\langle -1, 2, 7\rangle$ & 9 & 168 & F \\
65 & $\langle -1, 3\rangle$ & 11 & 168 & F \\
65 & $\langle -1, 6\rangle$ & 11 & 168 & F \\
65 & $\langle -1, 8, 12\rangle$ & 13 & 252 & E \\
65 & $\langle -1, 2\rangle$ & 19 & 336 & E \\
65 & $\langle -1, 7\rangle$ & 19 & 336 & E \\
65 & $\langle -1, 9\rangle$ & 21 & 336 & E \\
65 & $\langle -1, 8, 14\rangle$ & 25 & 504 & E \\
65 & $\langle -1, 12\rangle$ & 31 & 504 & E \\
65 & $\langle -1, 21\rangle$ & 31 & 504 & E \\
65 & $\langle -1, 4\rangle$ & 41 & 672 & E \\
65 & $\langle -1, 8\rangle$ & 55 & 1008 & E \\
65 & $\langle -1, 14\rangle$ & 61 & 1008 & E \\
65 & $\langle -1, 18\rangle$ & 55 & 1008 & E \\
67 & $\langle -1, 3\rangle$ & 15 & 204 & E \\
67 & $\langle -1, 29\rangle$ & 45 & 748 & E \\
68 & $\langle -1, 9\rangle$ & 13 & 216 & Z \\
72 & $\langle -1, 5\rangle$ & 13 & 288 & Z \\
72 & $\langle -1, 7\rangle$ & 13 & 288 & Z \\
72 & $\langle -1, 11\rangle$ & 9 & 288 & Z \\
72 & $\langle -1, 17, 19\rangle$ & 21 & 432 & Z \\
72 & $\langle -1, 23\rangle$ & 25 & 576 & Z \\
72 & $\langle -1, 17\rangle$ & 49 & 864 & Z \\
72 & $\langle -1, 19\rangle$ & 49 & 864 & Z \\
72 & $\langle -1, 35\rangle$ & 41 & 864 & Z \\
73 & $\langle -1, 6\rangle$ & 9 & 148 & D \\
73 & $\langle -1, 7\rangle$ & 13 & 222 & E \\
73 & $\langle -1, 2\rangle$ & 19 & 296 & E \\
73 & $\langle -1, 3\rangle$ & 25 & 444 & E \\
73 & $\langle -1, 10\rangle$ & 43 & 666 & E \\
73 & $\langle -1, 8\rangle$ & 55 & 888 & E \\
73 & $\langle -1, 27\rangle$ & 85 & 1332 & S \\
75 & $\langle -1, 4\rangle$ & 9 & 240 & T \\
75 & $\langle -1, 14\rangle$ & 17 & 480 & Z \\
75 & $\langle -1, 7\rangle$ & 37 & 600 & Z \\
75 & $\langle -1, 26\rangle$ & 73 & 1200 & S \\
80 & $\langle -1, 3, 13\rangle$ & 15 & 288 & Z \\
80 & $\langle -1, 7, 9\rangle$ & 15 & 288 & Z \\
80 & $\langle -1, 9, 11\rangle$ & 13 & 288 & Z \\
80 & $\langle -1, 7\rangle$ & 35 & 576 & Z \\
80 & $\langle -1, 13\rangle$ & 35 & 576 & Z \\
80 & $\langle -1, 17\rangle$ & 35 & 576 & Z \\
80 & $\langle -1, 9\rangle$ & 69 & 1152 & S \\
80 & $\langle -1, 31\rangle$ & 69 & 1152 & S \\
80 & $\langle -1, 39\rangle$ & 57 & 1152 & S \\
81 & $\langle -1, 8\rangle$ & 10 & 324 & Z \\
81 & $\langle -1, 26\rangle$ & 46 & 972 & Z \\
91 & $\langle -1, 3, 4\rangle$ & 13 & 224 & E \\
91 & $\langle -1, 2\rangle$ & 21 & 336 & E \\
91 & $\langle -1, 11\rangle$ & 21 & 336 & E \\
91 & $\langle -1, 5\rangle$ & 23 & 336 & E \\
91 & $\langle -1, 6\rangle$ & 23 & 336 & E \\
91 & $\langle -1, 9, 12\rangle$ & 25 & 448 & E \\
91 & $\langle -1, 3\rangle$ & 41 & 672 & E \\
91 & $\langle -1, 4\rangle$ & 41 & 672 & E \\
91 & $\langle -1, 25\rangle$ & 45 & 672 & E \\
91 & $\langle -1, 36\rangle$ & 45 & 672 & E \\
91 & $\langle -1, 8\rangle$ & 67 & 1008 & E \\
91 & $\langle -1, 9\rangle$ & 81 & 1344 & S \\
91 & $\langle -1, 12\rangle$ & 89 & 1344 & S \\
91 & $\langle -1, 16\rangle$ & 81 & 1344 & S \\
91 & $\langle -1, 22\rangle$ & 89 & 1344 & S \\
91 & $\langle -1, 27\rangle$ & 133 & 2016 & S \\
109 & $\langle -1, 3\rangle$ & 16 & 220 & E0 \\
109 & $\langle -1, 2\rangle$ & 22 & 330 & E0 \\
109 & $\langle -1, 4\rangle$ & 46 & 660 & E0 \\
109 & $\langle -1, 8\rangle$ & 64 & 990 & E0 \\
109 & $\langle -1, 45\rangle$ & 136 & 1980 & S \\
109 & $\langle -1, 33\rangle$ & 208 & 2970 & S \\
121 & $\langle -1, 10\rangle$ & 26 & 660 & E \\
121 & $\langle -1, 3\rangle$ & 106 & 1452 & S \\
125 & $\langle -1, 7\rangle$ & 36 & 750 & E0 \\
125 & $\langle -1, 24\rangle$ & 76 & 1500 & S \\
125 & $\langle -1, 57\rangle$ & 256 & 3750 & S \\
\bottomrule
\end{longtable}
\endgroup

\begin{table}[ht]\centering\small
\caption{Projections to intermediate curves (other than $X_0$) used in the proof of Theorem B.}\label{tab:projection}
\begin{tabular}{rlrl}
\toprule $N$ & Source generators & $M$ & Target generators \\
\midrule
63 & $\langle -1, 5\rangle$ & 63 & $\langle -1, 4, 5\rangle$ \\
68 & $\langle -1, 13\rangle$ & 34 & $\langle -1, 13\rangle$ \\
68 & $\langle -1, 33\rangle$ & 34 & $\langle -1, 13\rangle$ \\
80 & $\langle -1, 3\rangle$ & 40 & $\langle -1, 3\rangle$ \\
80 & $\langle -1, 9, 31\rangle$ & 40 & $\langle -1, 3\rangle$ \\
80 & $\langle -1, 11\rangle$ & 40 & $\langle -1, 11\rangle$ \\
80 & $\langle -1, 19\rangle$ & 40 & $\langle -1, 19\rangle$ \\
\bottomrule
\end{tabular}
\end{table}

\begin{table}[!ht]
\centering
\caption{The twelve exceptional negative certificates.}\label{tab:special}
\begin{tabular}{rrrl}
\toprule $N$ & Generators of $\Delta$ & $g$ & Method \\
\midrule
36 & $\langle -1, 17\rangle$ & 7 & Petri lifting \\
39 & $\langle -1, 5\rangle$ & 9 & Petri lifting \\
40 & $\langle -1, 3\rangle$ & 7 & Rational classes \\
43 & $\langle -1, 2\rangle$ & 9 & Empty cosets \\
48 & $\langle -1, 5\rangle$ & 7 & Rational classes \\
48 & $\langle -1, 7, 17\rangle$ & 7 & Rational classes \\
53 & $\langle -1, 4\rangle$ & 8 & Elliptic families \\
61 & $\langle -1, 4\rangle$ & 8 & Elliptic families \\
65 & $\langle -1, 2, 7\rangle$ & 9 & Elliptic families \\
65 & $\langle -1, 3\rangle$ & 11 & Elliptic families \\
65 & $\langle -1, 6\rangle$ & 11 & Elliptic families \\
75 & $\langle -1, 4\rangle$ & 9 & Petri lifting \\
\bottomrule
\end{tabular}
\end{table}

\begin{table}[ht]\centering\small
\caption{Theorem A, Route T: the known family $V=B+[F_4]+C$, the subgroups $S$ tested over $\Fp$, and the
counts. ``Divisors'' and ``classes'' are the effective divisors of degree $5$ over $\Fp$ and their classes.}
\label{tab:routeT}
\begin{tabular}{rrlrrlrrrr}\toprule
$N$&$p$&quotient by&$\dim B$&$\#B(\Fp)$&subgroups tested&divisors&classes&cosets&$\dim T Z_S$\\\midrule
74&3&$w_{74}$&2&21&all of order $7$&1340&1316&18&3\\
77&5&$w_{77}$&2&35&all of order $7$&12528&12488&40&3\\
82&3&$w_2,w_{41}$&1&6&$B(\F_3)$&2574&2568&6&2\\
86&7&$w_2,w_{43}$&1&8&cyclic of order $8$ ($192$)&64384&64384&12&2\\
99&5&$w_9,w_{11}$&1&10&$B(\F_5)$&8504&8504&6&2\\
111&2&$w_{111}$&2&15&all of order $5$&328&320&12&3\\
118&3&$w_2,w_{59}$&1&5&all of order $5$&3104&3104&6&2\\
123&2&$w_3,w_{41}$&1&3&order $3$ or $5$&842&842&6&2\\
128&3&$w_{128},sw_{128}s^{-1}$&1&6&$B(\F_3)$&1812&1812&4&2\\
141&2&$w_3,w_{47}$&1&3&order $3$ or $5$&566&566&4&2\\
142&3&$w_{71}$&2&15&order $5$ or $7$&1668&1656&12&3\\
143&2&$w_{143}$&2&15&all of order $3$&332&324&20&3\\
145&3&$w_5,w_{29}$&1&4&all of order $2$&1552&1552&24&2\\
155&2&$w_5,w_{31}$&1&3&order $3$ or $5$&536&536&4&2\\
159&5&$w_3,w_{53}$&1&6&all of order $3$&23476&23476&16&2\\\bottomrule
\end{tabular}
\end{table}

\begin{table}[ht]\centering\small
\caption{Theorem A, Route Q: $Y=X_0(N)/w_N$, $A=\pi^*\Jac Y$, $R=J/A$ with $R(\Q)$ cyclic of order $m$, and the
surviving indices $i$ of the translates $T_i=[5\infty]+it+A$.}\label{tab:routeQ}
\begin{tabular}{rrrrrrl}\toprule
$N$&$g$&$g(Y)$&$p$&$m$&$\#A(\Fp)$&surviving $i$\\\midrule
79&6&1&3&13&5&$1,2,3,4$\\
83&7&1&7&41&11&$1,2,3,4,23$\\
89&7&1&3&22&5&$1,2,3,4$\\
101&8&1&5&25&7&$1,2,3,4$\\
103&8&2&5&17&55&$2,3$\\
107&9&2&3&53&29&$2,3$\\
131&11&1&3&65&5&$1,2,3,4$\\
167&14&2&3&83&19&$2,3$\\
191&16&2&2&95&11&$2,3$\\\bottomrule
\end{tabular}
\end{table}
\begin{table}[ht]\centering\small
\caption{Theorem B, class E: for each level $N$ and index $t=[(\Z/N\Z)^\times:\Delta]$, the number of curves, their genera,
 the elliptic curves of conductor $N$ with odd analytic rank, and their optimal modular degree $d_E$ (Cremona's
 database). At $N=67,73$ there are none. Every positive-rank elliptic quotient has degree divisible by $t\,d_E$
 \cite[Corollary~4.10]{DerickxOrlic2026}, which is composite, hence $\ne5$.}\label{tab:classE}
\begin{tabular}{rrrlll}\toprule $N$&$t$&curves&genera&$E$&$d_E$\\\midrule
37&9&1&16&37a1&2\\
43&7&1&15&43a1&2\\
53&13&1&40&53a1&2\\
57&3&1&13&57a1&4\\
57&6&1&25&57a1&4\\
57&9&1&43&57a1&4\\
58&2&1&12&58a1&4\\
58&7&1&36&58a1&4\\
61&3&1&12&61a1&2\\
61&5&1&16&61a1&2\\
61&6&1&26&61a1&2\\
61&10&1&36&61a1&2\\
61&15&1&56&61a1&2\\
65&3&1&13&65a1&2\\
65&4&3&19, 19, 21&65a1&2\\
65&6&3&25, 31, 31&65a1&2\\
65&8&1&41&65a1&2\\
65&12&3&55, 55, 61&65a1&2\\
67&3&1&15&--&--\\
67&11&1&45&--&--\\
73&3&1&13&--&--\\
73&4&1&19&--&--\\
73&6&1&25&--&--\\
73&9&1&43&--&--\\
73&12&1&55&--&--\\
91&2&1&13&91a1, 91b1&4\\
91&3&4&21, 21, 23, 23&91a1, 91b1&4\\
91&4&1&25&91a1, 91b1&4\\
91&6&4&41, 41, 45, 45&91a1, 91b1&4\\
91&9&1&67&91a1, 91b1&4\\
121&5&1&26&121b1&4\\
\bottomrule\end{tabular}\end{table}

\end{document}