%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Formal immersion at the prime \texorpdfstring{$2$}{2}}\label{sec:formal}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section we show that the groups $\torz n$ for $n\in\{49,51,55,75,77,85,91,93\}$ are not subgroups of the
torsion of an elliptic curve over a number field of degree $10$. These levels were treated in degrees at most $9$ in
\cite[Sections 7 and 8]{N69}, and we follow the arguments given there. We reduce the trace divisor $D$ modulo $2$
and write its reduction as $D'+C'$ (\Cref{subsec:sieve}). The reductions with $D'\neq0$ are excluded with the Hecke
sieve, and for $n\in\{77,85,91,93\}$, where $J_1(n)$ has positive rank, with a variant of it which uses elements of the
Hecke algebra with finite image on the Mordell-Weil group. Except for $n\in\{91,93\}$ there are non-cuspidal closed
points of degree $10$ on $X_1(n)_{\F_2}$, which play no role in degree $9$. The entirely cuspidal reductions are
excluded with the formal immersion criterion of \Cref{prop:cartier}, except for $n=49$ and $n=75$, where
\Cref{prop:main}~b) applies, and $n=55$, where reduction modulo $2$ is injective on $J_1(55)(\Q)$. For $n=91$ and $n=93$
the argument of \cite[Proposition 8.5]{N69}, which uses \Cref{lem:stab}~c), would need $\gon_\C X_1(n)>2(10+10)=40$, which
we do not know. For $n=91$ we replace it by a relative version of \Cref{prop:cartier} (\Cref{lem:fi-relative}), and for $n=93$
we combine it with a descent to $X_1(31)$ and with \Cref{prop:cartier}.

%-------------------------------------------------------------------------------
\subsection{The curve \texorpdfstring{$X_1(49)$}{X1(49)}}\label{subsec:fi-49}

\begin{lemma}\label{lem:fi-49-points}
The non-cuspidal closed points of $X_1(49)_{\F_2}$ of degree at most $10$ are $42$ points of degree $9$, which form $2$
diamond orbits, and $126$ points of degree $10$, which form $6$ diamond orbits. For each of these points $P$ the
divisor $A_3P$ is not principal.
\end{lemma}

\begin{proof}
No elliptic curve over $\F_{2^e}$ with $e\leq 8$ has a point of order $49$ (see the proof of
\cite[Proposition 7.7]{N69}). We \gitlink{next_cases_20261030/cyclic_primepowers/pp49_finite_sieve.m}{compute} the
non-cuspidal closed points of degree $9$ and $10$ and their diamond orbits, and that $A_3P$ is not principal for a
representative $P$ of each of the $8$ orbits. As $A_3$ commutes with the diamond operators, this proves the lemma.
\end{proof}

\begin{proposition}\label{prop:49}
The group $\torz{49}$ is not a subgroup of the torsion of an elliptic curve over a number field of degree $10$.
\end{proposition}

\begin{proof}
Let $X=X_1(49)$, let $D$ be the trace divisor of a point of order $49$ over a number field of degree $10$, and let
$D'+C'$ be its reduction modulo $2$. The group $J_1(49)(\Q)$ is finite \cite[Theorem 3.1(2)]{DEHMZ}, and
$\gon_\Q X=21>20$ \cite[Theorem D]{Varivoda26}, so $D'\neq0$ by \Cref{prop:main}~b) with $p=2$. By
\Cref{lem:fi-49-points}, either $D'$ is a closed point of degree $10$ and $C'=0$, or $D'$ is a closed point of degree $9$
and $C'$ is a cusp of degree $1$. The cusps of degree $1$ of $X_{\F_2}$ are the reductions of the $21$ rational cusps of
$X$ (see the proof of \cite[Proposition 7.7]{N69}), so $3$ is admissible for $C'$, and the Hecke sieve (\Cref{subsec:sieve}) gives
$[A_3D']=0$, which contradicts \Cref{lem:fi-49-points}.
\end{proof}

%-------------------------------------------------------------------------------
\subsection{The curve \texorpdfstring{$X_1(51)$}{X1(51)}}\label{subsec:fi-51}

\begin{lemma}\label{lem:fi-51-points}
The non-cuspidal closed points of $X_1(51)_{\F_2}$ of degree at most $10$ are $80$ points of degree $9$, which form $5$
diamond orbits, and $64$ points of degree $10$, which form $4$ diamond orbits. For each of these points $P$ the
divisor $A_5P$ is not principal.
\end{lemma}

\begin{proof}
No elliptic curve over $\F_{2^e}$ with $e\leq8$ has a point of order $51$ (see the proof of
\cite[Proposition 7.9]{N69}). We \gitlink{next_cases_20261030/cyclic_further/C51/further51_finite_sieve.m}{compute} the
non-cuspidal closed points of degree $9$ and $10$ and their diamond orbits, and that $A_5P$ is not principal for a
representative $P$ of each of the $9$ orbits.
\end{proof}

\begin{lemma}\label{lem:fi-51-cusps}
Let $X=X_1(51)$ and $U=(\Cart-1)H^0(X_{\F_2},\Omega^1)$. The closed cusps of $X_{\F_2}$ have degrees $1^{16},2^8,8^4$
\cite[Proposition 7.9]{N69},
and the restriction map $U\to H^0(\overline C,\Omega^1|_{\overline C})$ is surjective for each of the
\[
[t^{10}](1-t)^{-16}(1-t^2)^{-8}(1-t^8)^{-4}=9656144
\]
effective cuspidal divisors $\overline C$ of degree $10$ on $X_{\F_2}$.
\end{lemma}

\begin{proof}
The curve $X$ has genus $65$, and $U$ has dimension $62$. In the proof of \cite[Proposition 7.9]{N69} we
\gitlink{next_cases_20261030/cyclic_further/C51/inputs/degree9/generic_cartier_completion_export.m}{computed} the
subspaces $\Omega(m\overline c)$ for the closed cusps $\overline c$ with $m\deg\overline c\leq8$. From them we
\gitlink{next_cases_20261030/cyclic_further/C51/further51_replay_old_jets.py}{check} that the restriction of $U$ is
surjective at each of the $9655880$ divisors $\sum_{\overline c}m_{\overline c}\overline c$ of degree $10$ with
$m_{\overline c}\deg\overline c\leq8$ for every $\overline c$. The remaining $264$ divisors are $10\overline c$ and
$9\overline c+\overline c'$ with distinct cusps $\overline c,\overline c'$ of degree $1$, and $5\overline r$ with
$\overline r$ of degree $2$. The diamond operator $\dm5$ generates $\Delta(51)$, it
\gitlink{next_cases_20261030/cyclic_further/C51/further51_cusp_orbits.py}{acts} transitively on the $16$ cusps of degree
$1$ and on the $8$ cusps of degree $2$, and $U$ is stable under it. So it suffices to consider $10\overline{c_0}$, the
$15$ divisors $9\overline{c_0}+\overline c'$ and $5\overline{r_0}$, where $\overline{c_0}$ is the reduction of the standard cusp
$c_0$ and $\overline{r_0}$ is a fixed cusp of degree $2$. We \gitlink{next_cases_20261030/cyclic_further/C51/further51_new_jets.m}{compute}
the subspaces $\Omega(10\overline{c_0})$ and $\Omega(5\overline{r_0})$, in a common basis of $H^0(X_{\F_2},\Omega^1)$ with
the subspaces above, and that the restriction of $U$ is surjective at these $17$ divisors.
\end{proof}

\begin{proposition}\label{prop:51}
The group $\torz{51}$ is not a subgroup of the torsion of an elliptic curve over a number field of degree $10$.
\end{proposition}

\begin{proof}
Let $X=X_1(51)$, let $D$ be the trace divisor of a point of order $51$ over a number field of degree $10$, and let
$D'+C'$ be its reduction modulo $2$. The group $J_1(51)(\Q)$ is finite \cite[Theorem 3.1(2)]{DEHMZ}.

Suppose first that $D'\neq0$. By \Cref{lem:fi-51-points}, either $D'$ is a closed point of degree $10$ and $C'=0$, or $D'$
is a closed point of degree $9$ and $C'$ is a cusp of degree $1$, which is the reduction of one of the $16$ rational
cusps of $X$. So $5$ is admissible for $C'$, and the Hecke sieve gives $[A_5D']=0$, which contradicts
\Cref{lem:fi-51-points}. So $D'=0$, and \Cref{prop:cartier} with $H=\{1\}$, $r=1$ and $\psi_1=1$, together with
\Cref{lem:fi-51-cusps}, gives a contradiction.
\end{proof}

%-------------------------------------------------------------------------------
\subsection{The curve \texorpdfstring{$X_1(55)$}{X1(55)}}\label{subsec:fi-55}

In degree $10$ the formal immersion criterion is not needed for $X_1(55)$, as the gonality is large enough once we
know that reduction modulo $2$ is injective on the rational torsion of the Jacobian. We use the quotient
$Y=X_1(55)/\dm{16}$ of \cite[Lemma 7.10]{N69}, which has genus $17$ and good reduction at $2$.

\begin{lemma}\label{lem:fi-55-inj}
Reduction modulo $2$ is injective on $J_1(55)(\Q)$.
\end{lemma}

\begin{proof}
The group $J_1(55)(\Q)$ is finite \cite[Theorem 3.1(2)]{DEHMZ}, and every torsion point in the kernel of reduction modulo
$2$ has order a power of $2$, so it suffices to consider $J_1(55)(\Q)[2^\infty]$, which is isomorphic to
$\torz2\times\torz{16}$ \cite[Theorem 4.13 and Table 2]{DEHMZ}. Let $\pi:X_1(55)\to Y$ be the quotient map. In the proof of
\cite[Proposition 7.11]{N69} we \gitlink{continuation_20261030b/C55/inputs/N55_quotient_injectivity_D9_v2.m}{computed}
that $J_Y(\F_2)\simeq(\torz5)^2\times\torz{10}\times\torz{1200}$ and that the reductions of the rational cuspidal divisors
generate a subgroup isomorphic to $\torz5\times\torz{10}\times\torz{1200}$, whose $2$-primary part has order $32$. The
group $J_Y(\Q)$ is finite, as $J_Y$ is a quotient of $J_1(55)$, and it contains the classes of the rational cuspidal
divisors, so the image of $J_Y(\Q)[2^\infty]$ under reduction has order at least $32$. As $\pi_*\pi^*=5$, the map $\pi^*$
is injective on $J_Y(\Q)[2^\infty]$, so this group has order at most $32$. Hence $J_Y(\Q)[2^\infty]$ has order $32$,
reduction modulo $2$ is injective on it, and $\pi^*$ maps it isomorphically onto $J_1(55)(\Q)[2^\infty]$. If $z=\pi^*y\in J_1(55)(\Q)[2^\infty]$ reduces to $0$, then so does $\pi_*z=5y$. So $5y=0$, and as
$y$ has order a power of $2$, we get $y=0$ and $z=0$.
\end{proof}

\begin{lemma}\label{lem:fi-55-points}
The non-cuspidal closed points of $X_1(55)_{\F_2}$ of degree at most $10$ are $20$ points of degree $7$, which form a
single diamond orbit, $40$ points of degree $9$, which form $2$ diamond orbits, and $100$ points of degree $10$, which
form $5$ diamond orbits. For each of these points $P$ the divisor $A_3P$ is not principal.
\end{lemma}

\begin{proof}
For the points of degree at most $9$ this is shown in the proof of \cite[Proposition 7.11]{N69}. We
\gitlink{computations/contb55_census.m}{compute} that exactly $50$ elliptic curves over $\F_{2^{10}}$ have a point of
order $55$, that their $j$-invariants form $5$ Frobenius orbits of length $10$, and that their subgroup of order $55$
is cyclic. This gives $100$ closed points of degree $10$, and the $20$ generators of the subgroup of order $55$ up
to sign are permuted transitively by the diamond operators, so they form $5$ diamond orbits. For a representative $P$ of
each of them we compute $T_3P$ from the $4$ isogenies of degree $3$ of an elliptic curve representing $P$. It consists
of a closed point of degree $10$ and one of degree $30$, or, for one orbit, of $4$ closed points of degree $10$, each
with multiplicity $1$, and we compute that $\ell(A_3P)=0$ (for the orbits
\gitlink{computations/contb55_source_case1.m}{1}, \gitlink{computations/contb55_source_case2.m}{2},
\gitlink{computations/contb55_source_case3.m}{3}, \gitlink{computations/contb55_source_case4.m}{4} and
\gitlink{computations/contb55_source_case5.m}{5}).
\end{proof}

\begin{proposition}\label{prop:55}
The group $\torz{55}$ is not a subgroup of the torsion of an elliptic curve over a number field of degree $10$.
\end{proposition}

\begin{proof}
Let $X=X_1(55)$, let $D$ be the trace divisor of a point of order $55$ over a number field of degree $10$, and let
$D'+C'$ be its reduction modulo $2$. The group $J_1(55)(\Q)$ is finite \cite[Theorem 3.1(2)]{DEHMZ}. The Galois orbits of
cusps of $X$ and the closed cusps of $X_{\F_2}$ both have degrees $1^{20},4^5,10^2,20$, and we
\gitlink{continuation_20261030b/C55/cusp_audit.py}{check} that every closed cusp of $X_{\F_2}$ is the reduction of a
Galois orbit of cusps of the same degree.

Suppose first that $D'\neq0$. By \Cref{lem:fi-55-points}, $D'$ is a closed point of degree $7$, $9$ or $10$, and $C'$
is a cuspidal divisor of degree $3$, $1$ or $0$, so it is supported on the cusps of degree $1$, which are the
reductions of the $20$ rational cusps. So $3$ is admissible for $C'$, and the Hecke sieve gives $[A_3D']=0$, which
contradicts \Cref{lem:fi-55-points}.

So $D'=0$, and $C'$ is the reduction of an effective rational cuspidal divisor $C$ of degree $10$. The class
$[D-C]\in J_1(55)(\Q)$ reduces to $0$, so $D\sim C$ by \Cref{lem:fi-55-inj}. As Abramovich's bound gives
$\gon_\Q X\geq\gon_\C X>14.28>10$, \Cref{lem:closed} gives $D=C$, which is a contradiction.
\end{proof}

%-------------------------------------------------------------------------------
\subsection{The curve \texorpdfstring{$X_1(75)$}{X1(75)}}\label{subsec:fi-75}

We use the quotient $Y=X_1(75)/\dm{26}$ of \cite[Section 7.6]{N69}, which has genus $73$ and good reduction at $2$, and
parametrizes elliptic curves with a point of order $25$ and a cyclic subgroup of order $3$. Its model over $\F_2$ is
obtained by adjoining to the function field of $X_1(25)_{\F_2}$ the common $x$-coordinate $x_3$ of the non-zero points of
a subgroup of order $3$ of the universal ordinary elliptic curve $y^2+xy=x^3+a_2x^2+1/j$, which satisfies
$x_3^4+x_3^3+1/j=0$. We write $\pi:X_1(75)\to Y$ for the quotient map, and $A_7=T_7-1-7\dm7$ also for the operator on $Y$.

\begin{lemma}\label{lem:fi-75-points}
The non-cuspidal closed points of $X_1(75)_{\F_2}$ of degree at most $10$ are $20$ points of degree $7$, which form a
single diamond orbit, and $120$ points of degree $10$, which form $6$ diamond orbits. For each of these points $P$ the
image $\pi(P)$ is a closed point of $Y_{\F_2}$ of the same degree as $P$, and the divisor $A_7\pi(P)$ on $Y_{\F_2}$ is not
principal.
\end{lemma}

\begin{proof}
For the points of degree at most $9$ this is shown in the proof of \cite[Proposition 7.13]{N69}. We
\gitlink{continuation_20261030b/C75/cont75_census10.m}{compute} that among the $2053$ elliptic curves over
$\F_{2^{10}}$ up to isomorphism exactly $60$ have a point of order $75$, that their $j$-invariants form $6$ Frobenius
orbits of length $10$, and that each of them has $E(\F_{2^{10}})\simeq\torz{1050}$. So each of them has a unique
subgroup of order $75$, whose $20$ generators up to sign are permuted transitively by the diamond operators, which gives
$120$ closed points of degree $10$ in $6$ diamond orbits. The degree of $\pi(P)$ divides that of $P$. For $P$ of degree
$10$ the $j$-invariant of $P$ generates $\F_{2^{10}}$, and for $P$ of degree $7$ the point $\pi(P)$ is not defined over
$\F_2$, as no elliptic curve over $\F_2$ has a point of order $25$. So in both cases $\pi(P)$ has the same degree as $P$.
For a representative
$P$ of each orbit of points of degree $10$ we compute $T_7\pi(P)$ from the $8$ isogenies of degree $7$ of an elliptic curve
representing $P$, carrying along the point of order $25$ and the subgroup of order $3$. It consists of $2$ closed points of
degree $10$ and $2$ of degree $30$, each with multiplicity $1$, and we compute that $\ell(A_7\pi(P))=0$ (for the orbits
\gitlink{continuation_20261030b/C75/cont75_nc10_job1_v2.m}{1 and 2}, \gitlink{continuation_20261030b/C75/cont75_nc10_job2_v2.m}{3
and 4}, and \gitlink{continuation_20261030b/C75/cont75_nc10_job3_v2.m}{5 and 6}). As $A_7$ commutes with the diamond
operators, this proves the lemma.
\end{proof}

\begin{proposition}\label{prop:75}
The group $\torz{75}$ is not a subgroup of the torsion of an elliptic curve over a number field of degree $10$.
\end{proposition}

\begin{proof}
Let $D$ be the trace divisor of a point of order $75$ over a number field of degree $10$, and let $D'+C'$ be its reduction
modulo $2$. We have $\rk J_1(75)(\Q)=0$ \cite[Theorem 3.1(2)]{DEHMZ}, and Abramovich's bound gives
$\gon_\Q X_1(75)>23.80>20$. So by \Cref{prop:main}~b) the reduction of $D$ is not entirely cuspidal, i.e.\ $D'\neq0$.

By \Cref{lem:fi-75-points}, $D'$ is a closed point $P$ of degree $7$ and $C'$ has degree $3$, or $P$ has degree $10$ and
$C'=0$. The closed cusps of $X_1(75)_{\F_2}$ of degree at most $3$ have degrees $1^{20},2^{10}$, and we
\gitlink{continuation_20261030b/C75/cont75_verify_cusps_v2.py}{check} that they are the reductions of Galois orbits of
cusps of $X_1(75)$ of the same degree. So $C'$ is the reduction of an effective rational cuspidal divisor $C$. The group
$J_Y(\Q)$ is finite, as $J_Y$ is a quotient of $J_1(75)$, so $A_7$ annihilates it by \Cref{prop:tools}~b), and
$A_7(\pi_*C)=A_7c_0=0$ by \Cref{prop:tools}~c). Hence $[A_7(\pi_*D-10c_0)]=0$, and its reduction gives
$[A_7\pi_*P]=0$. As $\pi_*P=\pi(P)$ by \Cref{lem:fi-75-points}, this contradicts \Cref{lem:fi-75-points}.
\end{proof}

%-------------------------------------------------------------------------------
\subsection{The curve \texorpdfstring{$X_1(77)$}{X1(77)}}\label{subsec:fi-77}

\begin{lemma}\label{lem:fi-77-Y}
Let $H\subseteq\Delta(77)$ be the subgroup of order $5$ generated by $\dm{36}$, let $Y=X_1(77)/H$, let
$\pi:X_1(77)\to Y$ be the quotient map, and let $\delta$ be the automorphism of $Y$ induced by $\dm2$.
\begin{itemize}
\item[a)] The curve $Y$ has genus $37$ and good reduction at $2$, and $Y\to X_0(77)$ is an unramified cyclic cover of
degree $6$, also over $\F_2$, whose group of automorphisms is generated by $\delta$. The automorphism of $Y$ induced by
$\dm3$ is $\delta^{-1}$.
\item[b)] The group $T_2J_Y(\Q)$ is finite.
\item[c)] The closed cusps of $X_1(77)_{\F_2}$ have degrees $1^{30},3^{10},10^3,30$. Those of degree $1$, $3$ and $10$ map to
the $6$ rational cusps of $Y_{\F_2}$ above the cusp of width $77$ of $X_0(77)$, to the $2$ closed cusps of degree $3$ above
the cusp of width $11$, and to the $3$ closed cusps of degree $2$ above the cusp of width $7$, respectively. In particular
the push-forward to $Y_{\F_2}$ of a closed cusp of degree $10$ is $5\overline r$ for a closed cusp $\overline r$ of degree $2$.
\end{itemize}
\end{lemma}

\begin{proof}
The first two assertions of a) are \cite[Lemma 7.15]{N69}. The group $\Delta(77)/H$ has order $6$, and as $2^6\equiv36^2$
and no smaller positive power of $2$ is congruent to $\pm36^i$ modulo $77$, it is generated by $\dm2$. As
$36^3\equiv-6\pmod{77}$, we have $\dm2\dm3=\dm6\in H$, which proves a). The group $T_2J_1(77)(\Q)$ is finite by
\cite[Lemma 7.14]{N69}. As $\pi_*$ commutes with $T_2$ and $\pi_*\pi^*=5$, we get $5T_2J_Y(\Q)\subseteq\pi_*T_2J_1(77)(\Q)$,
which proves b). The Frobenius elements at $2$ of the cusps of widths $77$, $11$, $7$ and $1$ of $X_0(77)$ in the
cover $Y\to X_0(77)$ are $\delta^0$, $\delta^2$, $\delta^3$ and $\delta^5$ (see \cite[Section 7.7]{N69}), of orders $1$,
$3$, $2$ and $6$, so above them there are $6$ rational cusps, $2$ closed cusps of degree $3$, $3$ of degree $2$ and one of
degree $6$. The cover $X_1(77)\to X_0(77)$ has degree $30$ and is unramified, so the closed cusps of degree $1$, $3$, $10$
and $30$ of $X_1(77)_{\F_2}$ lie above the cusps of widths $77$, $11$, $7$ and $1$, respectively, which gives c), as we
also \gitlink{continuation_20261030b/C77/cusp_audit.py}{check} with the explicit description of the cusps.
\end{proof}

\begin{lemma}\label{lem:fi-77-cusps}
Let $Y$ be as in \Cref{lem:fi-77-Y}, and let $U=\Cart(\Cart-1)H^0(Y_{\F_2},\Omega^1)$. Then $\dim U=32$, and the
restriction map $U\to H^0(\overline C,\Omega^1|_{\overline C})$ is surjective for each of the
\[
\sum_{r=0}^{3}(r+1)\tbinom{15-3r}{5}=4989
\]
effective divisors $\overline C$ of degree $10$ supported on the $6$ rational cusps and the $2$ closed cusps of degree $3$
of \Cref{lem:fi-77-Y}~c), and for the $3$ divisors $5\overline r$, where $\overline r$ is a closed cusp of degree $2$ of
\Cref{lem:fi-77-Y}~c).
\end{lemma}

\begin{proof}
We use the model of $Y_{\F_2}$ constructed after \cite[Lemma 7.15]{N69}, on which we
\gitlink{computations/contb77_fi10.m}{compute} the action of the automorphisms of $Y$ over $X_0(77)$ on
$H^0(Y_{\F_2},\Omega^1)$, which commutes with $\Cart$, and $\dim U=32$. We use the subspaces $\Omega(jP)$ for $j\leq8$ at
the rational cusps $P$ from the proof of \cite[Proposition 7.16]{N69}, and we compute the subspaces $\Omega(jP)$ for
$j\in\{9,10\}$ at a rational cusp, for $j\leq3$ at a cusp of degree $3$ and for $j=5$ at a cusp of degree $2$, which we
transport to the other cusps of the same degree with these automorphisms. From these we check that the restriction of $U$
is surjective at all the $4992$ divisors of the lemma.
\end{proof}

\begin{lemma}\label{lem:fi-77-points}
Let $Y$, $\pi$ and $\delta$ be as in \Cref{lem:fi-77-Y}. The non-cuspidal closed points of $X_1(77)_{\F_2}$ of degree at
most $10$ are $90$ points of degree $10$, which form $3$ diamond orbits. For each of these points $P$ the image
$Q=\pi(P)$ is a closed point of $Y_{\F_2}$ of degree $10$, and the divisor
\[
B=T+2\delta T-7Q-2\delta Q-3\delta^{-1}Q,\qquad T=T_3Q,
\]
on $Y_{\F_2}$ is not principal.
\end{lemma}

\begin{proof}
No elliptic curve over $\F_{2^e}$ with $e\leq9$ has a point of order $77$ (see the proof of \cite[Lemma 7.14]{N69}). We
\gitlink{computations/contb77_census.m}{compute} that exactly $30$ elliptic curves over $\F_{2^{10}}$ have a point of
order $77$, that each of them has $E(\F_{2^{10}})\simeq\torz{1078}$, and that their $j$-invariants form $3$ Frobenius
orbits of length $10$. The $30$ generators up to sign of the unique subgroup of order $77$ of each of them are permuted
transitively by the diamond operators, which gives $90$ closed points of degree $10$ in $3$ diamond orbits. As the
$j$-invariant of $P$ generates $\F_{2^{10}}$, the image $Q$ has degree $10$.

Our model of $Y_{\F_2}$ is a double cover of $X_1(7)\times_{X(1)}X_0(11)$, given by an Artin-Schreier equation, and its
automorphism over this curve is $\delta^3$. We compute the images on $X_1(7)\times_{X(1)}X_0(11)$ of $P$ and of the targets
of the $4$ isogenies of degree $3$ of an elliptic curve representing $P$, from the Tate normal form of the point of order
$7$ and the $j$-invariants of the curve and of its quotient by the subgroup of order $11$. So each closed point occurring in
$T$ is determined up to $\delta^3$. Replacing $Q$ by $\delta^3Q$ replaces $B$ by $\delta^3B$, so we may fix $Q$, and we test
all the possible choices of the closed points in $T$. For the representatives of the $3$ orbits the kernels of the
isogenies of degree $3$ form Frobenius orbits of lengths $(1,3)$, $(1,1,1,1)$ and $(1,3)$, the corresponding closed points
have degrees $10$ and $30$, each with multiplicity $1$, and there are $4$, $16$ and $4$ choices. We compute that
$\ell(B)=0$ in all these $24$ cases (for the orbits \gitlink{computations/contb77_nc_case1_v3.m}{1},
\gitlink{computations/contb77_nc_case2_v3.m}{2} and \gitlink{computations/contb77_nc_case3_v3.m}{3}). As $T_3$ commutes with
the diamond operators, this proves the lemma.
\end{proof}

\begin{proposition}\label{prop:77}
The group $\torz{77}$ is not a subgroup of the torsion of an elliptic curve over a number field of degree $10$.
\end{proposition}

\begin{proof}
Let $D$ be the trace divisor of a point of order $77$ over a number field of degree $10$, let $D'+C'$ be its reduction
modulo $2$, and let $Y$, $\pi$ and $\delta$ be as in \Cref{lem:fi-77-Y}.

Suppose first that $D'=0$. As $C'$ has degree $10$, it is supported on the closed cusps of degree $1$ and $3$, or it is a
single closed cusp of degree $10$. By \Cref{lem:fi-77-Y}~c), the reduction of $\pi_*D$ is then one of the divisors of
\Cref{lem:fi-77-cusps}. The group $T_2J_Y(\Q)$ is finite by \Cref{lem:fi-77-Y}~b), and we apply \Cref{prop:cartier} with
$r=1$ and $\psi_1=T_2$. By the Eichler-Shimura relation $T_2^*=\Cart\dm2^*$ on $H^0(Y_{\F_2},\Omega^1)$, and as $\dm2^*$
is invertible and commutes with $\Cart$, we get
\[
(\Cart-1)T_2^*H^0(Y_{\F_2},\Omega^1)=\Cart(\Cart-1)H^0(Y_{\F_2},\Omega^1).
\]
So \Cref{lem:fi-77-cusps} and \Cref{prop:cartier} give a contradiction.

So $D'\neq0$, and by \Cref{lem:fi-77-points} the divisor $D'$ is a closed point $P$ of degree $10$ and $C'=0$, and the
reduction of $\pi_*D$ is $Q=\pi(P)$. As $T_2J_Y(\Q)$ is finite, the operator $A_3T_2$ annihilates $J_Y(\Q)$ by
\Cref{prop:tools}~b). By the Eichler-Shimura relation $T_2=F+\delta V$ on $J_{Y,\F_2}$, and as the Frobenius $F$ is the
identity on $J_Y(\F_2)$ and $VF=2$, the operator $T_2$ acts on $J_Y(\F_2)$ as $1+2\delta$. By \Cref{lem:cuspHecke} we have
$A_3c_0=A_3\delta c_0=0$ as divisors. So reducing $A_3T_2[\pi_*D-10c_0]=0$ modulo $2$ we obtain $[(1+2\delta)A_3Q]=0$. As
$\dm3$ acts on $Y$ as $\delta^{-1}$, we have $(1+2\delta)A_3Q=B$, with $B$ as in \Cref{lem:fi-77-points}, which is a
contradiction.
\end{proof}

%-------------------------------------------------------------------------------
\subsection{The curve \texorpdfstring{$X_1(85)$}{X1(85)}}\label{subsec:fi-85}

For the levels $85$ and $93$ we compute, as in \cite[Section 8]{N69}, with the model of $X_1(n)$ given by $q$-expansions
(\Cref{subsec:hecke}). The reductions modulo $2$ of a basis of the lattice of cusp forms of weight $2$ on $\Gamma_1(n)$ with
integral $q$-expansions form a basis of $H^0(X_1(n)_{\F_2},\Omega^1)$, and we compute the Hecke and diamond operators on
it. On $q$-expansions with coefficients in $\F_2$ the Cartier operator acts as Atkin's operator
$\sum_ja_jq^j\mapsto\sum_ja_{2j}q^j$ (see \cite[Section 7]{N69}), which is also the reduction modulo $2$ of $T_2$, as $n$ is
odd. At each rational cusp $c$ we use the local parameter obtained from the parameter $q$ at the cusp $\infty$ with a
diamond operator, and for $j\geq1$ we write $\lambda_{c,j}(\omega)$ for the $j$-th coefficient of the expansion of a
differential $\omega$ at $c$, i.e.\ $\omega=\sum_{j\geq1}\lambda_{c,j}(\omega)q^{j-1}dq$ near $c$. If
$\overline C=\sum_cm_c\overline c$ is supported on the rational cusps, then the restriction of a subspace $U$ of
$H^0(X_1(n)_{\F_2},\Omega^1)$ to $\overline C$ is surjective if and only if the linear forms $\lambda_{c,j}$ with
$1\leq j\leq m_c$ are linearly independent on $U$.

\begin{lemma}\label{lem:fi-85-MW}
Put $k_2(X)=(X^2+2X-1)(X^2+2X+1)$ and $k_3(X)=(X^2+4X+2)(X^2+2X+2)$. Then the groups $(\dm2-1)J_1(85)(\Q)$,
$k_2(T_2)J_1(85)(\Q)$ and $k_3(T_3)J_1(85)(\Q)$ are finite.
\end{lemma}

\begin{proof}
By the proof of \cite[Proposition 8.5]{N69}, the factors of $J_1(85)$ outside the winding quotient are $2$ factors of
dimension $2$, whose characters are trivial on $\dm2$, and all the other factors have finitely many rational points. This
gives the first claim. We
\gitlink{continuation_20261030c/C85/inherited_degree9/prime_orders/continued_20261026/composite85/rational_cusp_FI_positive.m}{compute}
that the characteristic polynomials of $T_2$ on these $2$ factors are $X^2+2X-1$ and $X^2+2X+1$, and those of $T_3$ are
$X^2+4X+2$ and $X^2+2X+2$. So $k_2(T_2)$ and $k_3(T_3)$ act as $0$ on these factors, which proves the other claims.
\end{proof}

\begin{lemma}\label{lem:fi-85-cusps}
Let $X=X_1(85)$ and
\[
U=(\Cart-1)\bigl(k_2(T_2)^*H^0(X_{\F_2},\Omega^1)+k_3(T_3)^*H^0(X_{\F_2},\Omega^1)\bigr),
\]
with $k_2,k_3$ as in \Cref{lem:fi-85-MW}. The closed cusps of $X_{\F_2}$ have degrees $1^{32},4^8,8^8$
\cite[Proposition 8.5]{N69}, and the restriction
map $U\to H^0(\overline C,\Omega^1|_{\overline C})$ is surjective for every effective cuspidal divisor $\overline C$ of
degree $10$ on $X_{\F_2}$.
\end{lemma}

\begin{proof}
As $k_2(X)\equiv X^4+1$ and $k_3(X)\equiv X^4\pmod 2$, the space $U$ is spanned by the images of $(T_2-1)(T_2^4+1)$ and
$(T_2-1)T_3^4$ on the reductions of the $q$-expansions, and we
\gitlink{continuation_20261030c/C85/contc85_cusp_blocks.py}{compute} that it has dimension $198$. It is stable under the
diamond operators, as they commute with $\Cart$ and with the Hecke operators, so it suffices to prove the claim for one
divisor in each diamond orbit.

Suppose first that $\overline C=\sum_cm_c\overline c$ is supported on the $32$ rational cusps. The diamond operators act
transitively on them, so we may assume that $m_\infty$ is the largest multiplicity, and then $m_c\leq5$ for $c\neq\infty$.
We compute the $165$ linear forms $\lambda_{\infty,j}$ with $j\leq10$
and $\lambda_{c,j}$ with $j\leq5$ for the other $31$ rational cusps $c$ on $U$. They span a space of dimension $152$. For a
linear relation between them let $j_c$ be the largest $j$ for which $\lambda_{c,j}$ occurs in it, with $j_c=0$ if there is
none, and put $w=\max_cj_c+\sum_{c\neq\infty}j_c$. We \gitlink{continuation_20261030c/C85/contc85_cusp_FI.py}{check}
that $w\geq32$ for each of the $2^{13}-1=8191$ non-trivial linear relations. A relation between the forms $\lambda_{c,j}$
with $j\leq m_c$ would have $j_c\leq m_c\leq m_\infty$ for every $c$, and hence $w\leq m_\infty+\sum_{c\neq\infty}m_c=10$. So
these forms are linearly independent on $U$.

Suppose now that $\overline C$ contains a cusp of degree $4$ or $8$. The diamond operators act transitively on the $8$
cusps of degree $4$, and the $8$ cusps of degree $8$ form $2$ diamond orbits. So we may assume that $\overline C$ is the
sum of a fixed cusp of degree $4$ and a divisor of degree $6$ on the rational cusps, or of a fixed cusp of degree $4$, one
of the $8$ cusps of degree $4$ and a divisor of degree $2$ on the rational cusps, or of one of $2$ fixed cusps of degree
$8$ and a divisor of degree $2$ on the rational cusps. We compute the expansions of the differentials at these cusps, to
$2$ coefficients at the cusps of degree $4$ and to $1$ coefficient at the cusps of degree $8$, from the Tate curves
$E(q^{17})$ with the point $\zeta_5q$ (\gitlink{continuation_20261030c/C85/contc85_arc_q4.m}{degree $4$}), $E(q^5)$ with
the point $\zeta_{17}q$ (\gitlink{continuation_20261030c/C85/contc85_arc_q8width5.m}{degree $8$}) and $E(q)$ with the
point $\zeta_{85}$ (\gitlink{continuation_20261030c/C85/contc85_arc_q8width1.m}{degree $8$}), using the modular units
of \cite[Sections 2 and 2.1]{DvH}, and we \gitlink{continuation_20261030c/C85/contc85_cusp_FI.py}{check} that the
restriction of $U$ is surjective at each of the
\[
\tbinom{37}{6}+8\tbinom{33}{2}+2\tbinom{33}{2}=2330064
\]
divisors of these forms.
\end{proof}

\begin{lemma}\label{lem:fi-85-points}
The non-cuspidal closed points of $X_1(85)_{\F_2}$ of degree at most $10$ are $160$ points of degree $9$, which form $5$
diamond orbits, and $128$ points of degree $10$, which form $4$ diamond orbits. Put $\sigma=\dm2^{-1}$, $\tau=\dm3$ and
$A_3=T_3-1-3\tau$. For each of these points $P$ the divisor
\[
B=A_3(\sigma P-P)=\sigma T-T-\sigma P+P-3\sigma\tau P+3\tau P,\qquad T=T_3P,
\]
is not principal.
\end{lemma}

\begin{proof}
For the points of degree at most $9$ this is shown in the proof of \cite[Proposition 8.5]{N69}. We
\gitlink{computations/contc_parent_source_census.py}{compute} that the non-cuspidal closed points of degree $10$ come from
elliptic curves over $\F_{2^{10}}$ with $1020$ points, whose $j$-invariants form $4$ Frobenius orbits of length $10$ and
whose groups of points are cyclic. This gives $128$ closed points of degree $10$, which form $4$ diamond orbits of length
$32$. For a representative $P$ of each orbit, $T_3P$ consists of a closed point of degree $10$ and one of degree $30$, each
with multiplicity $1$, so $B$ has positive and negative parts of degree $80$. As in \cite[Section 8]{N69} we
\gitlink{continuation_20261030c/C85/contc85_jets10.m}{compute} local parameters at $P$ and at the points of $T_3P$ from the
deformation $a_6\mapsto a_6+t$ of the corresponding ordinary elliptic curves, and their images on the model given by
$q$-expansions, of genus $225$, from the modular units of \cite[Sections 2 and 2.1]{DvH}. On this model we compute $\sigma
P$, $\tau P$ and $\sigma\tau P$ with the diamond operators, using that the diamond operator $\dm a$ of this model
corresponds to $\dm a^{-1}$ on the moduli problem, which we
\gitlink{continuation_20261030c/C85/inherited_degree9/prime_orders/continued_20261026/composite85/verify_stage1_readonly.py}{verify}
for $a\in\{2,3\}$ by comparing the modular units at the points $q$ and $q^a$ of order $85$ of the Tate curve $E(q^{85})$. We
\gitlink{continuation_20261030c/C85/contc85_R1_sieve.py}{check} with \Cref{lem:mult}, with $\mathcal L=\mathcal M=K_X$,
for which $H^0(X,2K_X)$ has dimension $672$, that $\ell(B)=0$ for these $4$ representatives. As $A_3$ commutes with the
diamond operators, this proves the lemma.
\end{proof}

\begin{proposition}\label{prop:85}
The group $\torz{85}$ is not a subgroup of the torsion of an elliptic curve over a number field of degree $10$.
\end{proposition}

\begin{proof}
Let $D$ be the trace divisor of a point of order $85$ over a number field of degree $10$, and let $D'+C'$ be its reduction
modulo $2$.

Suppose first that $D'=0$. By \Cref{lem:fi-85-MW} we may apply \Cref{prop:cartier} with $H=\{1\}$, $r=2$,
$\psi_1=k_2(T_2)$ and $\psi_2=k_3(T_3)$, and by \Cref{lem:fi-85-cusps} it shows that $D$ does not reduce to $C'$, which is a
contradiction.

So $D'\neq0$, and by \Cref{lem:fi-85-points}, either $D'$ is a closed point $P$ of degree $10$ and $C'=0$, or $D'$ is a
closed point $P$ of degree $9$ and $C'$ is a cusp of degree $1$, which is the reduction of a rational cusp $c$. Let $\sigma$,
$\tau$ and $A_3$ be as in \Cref{lem:fi-85-points}. The class
$[\sigma D-D]=(\sigma-1)[D-10c_0]+10[\sigma c_0-c_0]$ is torsion by \Cref{lem:fi-85-MW} and by the Manin-Drinfeld theorem, so it
is annihilated by $A_3$ by \Cref{prop:tools}~b). As $A_3$ annihilates the rational cusps $c$ and $\sigma c$ as divisors, the
reduction of $A_3(\sigma D-D)$ is $A_3(\sigma P-P)$, so $A_3(\sigma P-P)$ is principal. This contradicts
\Cref{lem:fi-85-points}.
\end{proof}

%-------------------------------------------------------------------------------
\subsection{The curve \texorpdfstring{$X_1(91)$}{X1(91)}}\label{subsec:fi-91}

For $n=91$ we work on a quotient $Y$ of $X_1(91)$ of genus $23$ with an automorphism $\tau$ of order $3$, and use the
element $\tau-1$, which has finite image on $J_Y(\Q)$. As it kills the pull-backs of divisors from $Y/\dm\tau$, the
criterion of \Cref{prop:cartier} fails at the cuspidal divisors which contain such pull-backs. The following lemma shows
that near such a divisor the fibre of the corresponding map to the Jacobian consists only of divisors of this form.

\begin{lemma}\label{lem:fi-relative}
Let $Y=X_1(n)/H$ be as in \Cref{subsec:formal}, let $\tau$ be an automorphism of $Y$ of order $3$ induced by a diamond
operator, let $Z=Y/\dm\tau$, assume that $Z$ has positive genus, and let $\pi:Y\to Z$ be the quotient map. Assume that
$(\tau-1)J_Y(\Q)$ is finite, and put
\[
U=(\Cart-1)(\tau^*-1)H^0(Y_{\F_2},\Omega^1).
\]
Let $A$ be an effective rational divisor of degree $d$ on $Y$, and let $C$ be an effective divisor of degree $d$ on $Y$
defined over $\Q_2$, with the same reduction $\overline C$ modulo $2$ as $A$, such that $(\tau-1)[C-dc_0]$ is torsion.
Suppose that $C=C_0+\pi^*E$ for an effective divisor $E$ of degree $k$ on $Z$ defined over $\Q_2$, such that the morphism
$Z^{(k)}\to Y^{(d)}$, $E'\mapsto C_0+\pi^*E'$, of the smooth models over $\Z_2$ induces a surjection of the completed local
rings at $\overline E$ and $\overline C$. If the restriction map $U\to H^0(\overline C,\Omega^1|_{\overline C})$ has rank
$d-k$, then $A\geq C_0$.
\end{lemma}

\begin{proof}
By \Cref{lem:goodred} the curve $Z$ has good reduction at $2$ and $\pi$ extends to the smooth models over $\Z_2$. Put
$\alpha=\tau-1$ and $M=T_2-\dm2$. The class $\alpha[A-C]=\alpha[A-dc_0]-\alpha[C-dc_0]\in J_Y(\Q_2)$ is torsion by
assumption, and it reduces to $0$, as $A$ and $C$ have the same reduction. It is killed by $2$, by the argument behind the
result of Parent recalled in \Cref{subsec:hecke} (see \cite[Section 2.3]{N69}), which is local at $2$. So $M\alpha[A-C]=0$
by \Cref{lem:dyadic}, applied to the N\'eron model of $J_Y$ over $\Z_2$ with $T=T_2$ and $\delta=\dm2$. Consider the
morphism
\[
\Psi:Y^{(d)}\to J_Y,\qquad B\mapsto M\alpha[B-dc_0]
\]
over $\Z_2$. Then $\Psi(A)=\Psi(C)$. As in the proof of \cite[Proposition 7.3]{N69}, the image of the cotangent map of
$\Psi$ at $\overline C$ is the restriction to $\overline C$ of $(\tau^*-1)(\Cart-1)\dm2^*H^0(Y_{\F_2},\Omega^1)$, which equals
$U$, as $\dm2^*$ is invertible and commutes with $\Cart$ and $\tau^*$. By assumption this restriction has dimension $d-k$.
Choose local coordinates $f_1,\dots,f_g$ of $J_Y$ over $\Z_2$ at $\Psi(C)$, vanishing at $\Psi(C)$, such that the
differentials of $f_1\circ\Psi,\dots,f_{d-k}\circ\Psi$ at $\overline C$ are linearly independent. As $Y^{(d)}$ is smooth
over $\Z_2$ of relative dimension $d$, the closed formal subscheme $\mathcal V$ of the formal neighbourhood of $\overline C$
defined by $f_1\circ\Psi,\dots,f_{d-k}\circ\Psi$ is formally smooth over $\Z_2$ of relative dimension $k$, i.e.\ its ring is
isomorphic to $\Z_2[[s_1,\dots,s_k]]$. As $\tau\pi^*=\pi^*$ on $J_Z$, we have
$\Psi(C_0+\pi^*E')=\Psi(C)+M\alpha\pi^*[E'-E]=\Psi(C)$ for every $E'$, so the image of the formal neighbourhood of
$\overline E$ in $Z^{(k)}$, whose ring is isomorphic to $\Z_2[[e_1,\dots,e_k]]$, lies in $\mathcal V$. By assumption the
corresponding homomorphism $\Z_2[[s_1,\dots,s_k]]\to\Z_2[[e_1,\dots,e_k]]$ is surjective, and as both rings are integral
domains of dimension $k+1$, it is an isomorphism. So $\mathcal V$ is the image of the formal neighbourhood of $\overline E$.
The point $A\in Y^{(d)}(\Z_2)$ reduces to $\overline C$ and $\Psi(A)=\Psi(C)$, so it lies in $\mathcal V$, and hence
$A=C_0+\pi^*E'$ for some $E'\in Z^{(k)}(\Z_2)$. In particular $A\geq C_0$.
\end{proof}

Note that for $k=0$ the conclusion is $A=C$, as in \Cref{prop:cartier} with $r=1$ and $\psi_1=\tau-1$.

\begin{lemma}\label{lem:fi-91-Y}
Let $H\subseteq\Delta(91)$ be the subgroup of the $\dm a$ with $a\equiv\pm1\pmod 7$, which has order $12$, let
$Y=X_1(91)/H$, let $\pi_X:X_1(91)\to Y$ be the quotient map, let $\tau$ be the automorphism of $Y$ induced by $\dm3$, and let
$\pi:Y\to Z=Y/\dm\tau$ be the quotient map.
\begin{itemize}
\item[a)] The group $H$ acts freely on the geometric points of $X_1(91)$, in characteristic $0$ and $2$. The curve $Y$ is the
normalization of $X_1(7)\times_{X(1)}X_0(13)$, with $\pi_X(E,P)=(E,13P,\dm{7P})$, it has genus $23$ and good reduction
at $2$, and it is a curve as in \Cref{subsec:formal}. The automorphism $\tau$ has order $3$, and $Z=X_0(91)$ has genus $7$.
The group $(\tau-1)J_Y(\Q)$ is finite.
\item[b)] The closed cusps of $X_1(91)_{\F_2}$ have degrees $1^{36},3^{12},12^6$. The push-forward by $\pi_X$ maps each of
those of degree $1$ to one of the reductions $\overline{c_1},\overline{c_2},\overline{c_3}$ of the $3$ rational cusps
$c_1,c_2,c_3$ of $Y$ of width $91$, and each of those of degree $3$ to the reduction $\overline q$, which has degree $3$, of a closed
cusp $q$ of $Y$ of degree $3$ and width $13$, in both cases with multiplicity $1$. The automorphism $\tau$ permutes $c_1,c_2,c_3$
cyclically and the $3$ geometric points of $q$ transitively, and $\pi$ is unramified at these cusps.
\item[c)] The fixed points of $\tau$ on $Y_{\F_2}$ are $2$ closed points $p_1,p_2$ of degree $2$, and $\tau$ acts on the
tangent spaces at them by a primitive cube root of unity. Each $p_i$ is the reduction of a unique point $P_i$ of $Y$ of
degree $2$ over $\Q_2$ with $\tau P_i=P_i$, and $\pi^*\pi(P_i)=3P_i$.
\item[d)] The non-cuspidal closed points of $X_1(91)_{\F_2}$ of degree at most $10$ are $3$ points of degree $8$ and $144$
points of degree $9$. The push-forward by $\pi_X$ of each of the points of degree $8$ is $4p_1$ or $4p_2$.
\end{itemize}
\end{lemma}

\begin{proof}
a) If $\dm a\in H$ fixes a geometric point, then by \cite[Lemma 7.2]{N69} either it is a cusp and $a$ is congruent to the
same sign $\pm1$ modulo $7$ and $13$, or it is not a cusp and $a\not\equiv\pm1\pmod 7$. As $a\equiv\pm1\pmod 7$, in both cases
$\dm a=1$. The map $\pi_X$ is invariant under $H$, and $X_1(91)\to X_1(7)\times_{X(1)}X_0(13)$ has degree
$i(91)/(i(7)\cdot14)=12$, which proves the second assertion. By the Riemann-Hurwitz formula $Y$ has genus
$1+(265-1)/12=23$, as $X_1(91)$ has genus $265$, and as $H$ acts freely also in characteristic $2$, it is a curve as in
\Cref{subsec:formal}, with good reduction at $2$. As $3$ has order $6$
in $(\torz7)^\times$, the automorphism $\tau$ generates $\Delta(91)/H$, which has order $3$, and $Z=X_1(91)/\Delta(91)=X_0(91)$.
The group $(\dm3-1)J_1(91)(\Q)$ is finite by the proof of \cite[Proposition 8.5]{N69}. As $\pi_{X*}$ intertwines $\dm3$ and
$\tau$ and $\pi_{X*}\pi_X^*=12$, we get $12(\tau-1)J_Y(\Q)\subseteq\pi_{X*}(\dm3-1)J_1(91)(\Q)$, so $(\tau-1)J_Y(\Q)$ is
finite.

We construct $Y_{\F_2}$ as follows. Over $\F_2(t)$ put $b=t^3-t^2$, $c=t^2-t$ and
$E_t:y^2+(1-c)xy-by=x^3-bx^2$, on which $(0,0)$ has order $7$, and
\[
j_{13}(s)=\frac{(s^2+5s+13)(s^4+7s^3+20s^2+19s+1)^3}{s}.
\]
We \gitlink{continuation_20261030c/C91/tate7_quotient_common.m}{check} that $j_{13}(s)=j(E_t)$ defines an irreducible
extension of degree $14$ of $\F_2(t)$ of genus $23$, so its normalization is $Y_{\F_2}$, and that $t\mapsto1/(t+1)$,
$s\mapsto s$ defines an automorphism of $Y_{\F_2}$ of order $3$ over $X_0(91)_{\F_2}$. It acts on $X_1(7)$ as a diamond operator and trivially on $X_0(13)$, so it is $\tau$ or $\tau^{-1}$, and as
all our assertions depend only on the group generated by $\tau$, we may compute with it.

b) The cusps of $Y_{\F_2}$ are the
\gitlink{computations/contc91_tate7_cusps.m}{poles} of $j(E_t)$, namely $3$ rational cusps of width $91$, $3$ of width $7$,
and one closed cusp of degree $3$ of width $13$ and one of width $1$, and the Galois orbits of cusps of $Y$ have the same
degrees. The degrees of the closed cusps of $X_1(91)_{\F_2}$ and the other assertions follow from the description of the
cusps, as we \gitlink{continuation_20261030c/C91/tate7_cusp_audit_v2.py}{check}.

c) The fixed points of $\tau$ on $Y_{\F_2}$ are the ramification points of $Y_{\F_2}\to Z_{\F_2}$, which is tamely ramified
of degree $3$. As $Y_{\F_2}$ and $Z_{\F_2}$ have genus $23$ and $7$, the Riemann-Hurwitz formula
$44=3\cdot12+2r$ shows that there are $r=4$ geometric fixed points. We
\gitlink{computations/contc91_export_native_v2.m}{compute} that above the closed point $t^2+t+1=0$ there are $2$ closed
points of degree $2$ of $Y_{\F_2}$ which are fixed by $\tau$, and that $\tau$ multiplies a uniformizer at each of them by a
primitive cube root of unity, modulo its square. These are the $4$ geometric fixed points. So the fixed point scheme of
$\tau$ over $\Z_2$ is \'etale at $p_i$, which gives the unique lift $P_i$, and $\pi$ is totally ramified at $P_i$.

d) The non-cuspidal closed points of degree at most $9$ are given in \cite[Proposition 8.5]{N69}, and we
\gitlink{computations/contc_parent_source_census.py}{compute} that there are none of degree $10$. The points of degree $8$
have $j=0$ and come from $2$ elliptic curves over $\F_{2^8}$ with $273$ points \cite[Proposition 8.5]{N69}. We
\gitlink{computations/contc_parent_source_census.py}{compute} that their groups of points are cyclic, and we
\gitlink{continuation_20261030c/C91_CM_control/CM_level7_binding.m}{check} that on each
of them an automorphism $u$ of order $3$ satisfies $u(P)=16P$ for a point $P$ of order $91$, and hence for every point of
order $91$ over $\F_{2^8}$, as these are multiples of $P$. As
$16\equiv2\pmod7$ and $16\equiv3\pmod{13}$, the automorphism $u$ gives an isomorphism between $(E,13P,\dm{7P})$ and
$(E,2\cdot13P,\dm{7P})$, which is the image of $(E,13P,\dm{7P})$ under the automorphism of $Y$ induced by $\dm2$, and this
automorphism is $\tau^{-1}$, as $2\equiv3^2\pmod 7$. So the image of a point of degree $8$ on $Y_{\F_2}$ is fixed by $\tau$,
and by c) it is $p_1$ or $p_2$. As it has degree $2$, the push-forward is $4p_1$ or $4p_2$.
\end{proof}

\begin{lemma}\label{lem:fi-91-ranks}
Let $Y$, $\tau$, $c_1,c_2,c_3$, $q$ and $p_1,p_2$ be as in \Cref{lem:fi-91-Y}, and let
$U=(\Cart-1)(\tau^*-1)H^0(Y_{\F_2},\Omega^1)$. Then $\dim U=14$, and the following hold.
\begin{itemize}
\item[a)] For each of the $120$ divisors $\overline C=a_1\overline{c_1}+a_2\overline{c_2}+a_3\overline{c_3}+b\overline q$ with
$a_1+a_2+a_3+3b=10$, the restriction map $U\to H^0(\overline C,\Omega^1|_{\overline C})$ has rank
$10-\min(a_1,a_2,a_3)-b$.
\item[b)] For $i\in\{1,2\}$ and each of the $6$ effective divisors $\overline R$ of degree $2$ supported on
$\overline{c_1},\overline{c_2},\overline{c_3}$, the restriction map $U\to H^0(\overline C,\Omega^1|_{\overline C})$ for
$\overline C=4p_i+\overline R$ has rank $8$.
\end{itemize}
\end{lemma}

\begin{proof}
On the model of the proof of \Cref{lem:fi-91-Y} we compute the matrices of $\Cart$ and $\tau^*$, which commute, and of
ranks $21$ and $16$ for $\Cart$ and $\tau^*-1$, and $\dim U=14$. We compute the spaces $\Omega(\overline C)$ and the ranks for
\gitlink{computations/contc91_tate7_cusps.m}{a)} and \gitlink{computations/contc91_tate7_CM.m}{b)}.
\end{proof}

\begin{lemma}\label{lem:fi-91-immersion}
Let $Y$, $Z$, $\pi$, $c_1,c_2,c_3$, $q$ and $P_1,P_2$ be as in \Cref{lem:fi-91-Y}, considered over $\Z_2$, and let $C=C_0+\pi^*E$ be
one of the following divisors.
\begin{itemize}
\item[a)] $C=a_1c_1+a_2c_2+a_3c_3+bq$ with $a_1+a_2+a_3+3b=10$, $E=m\pi(c_1)+b\pi(q)$ with $m=\min(a_1,a_2,a_3)$, and
$C_0=C-\pi^*E$.
\item[b)] $C=4P_i+R$ with $R$ an effective divisor of degree $2$ supported on $c_1,c_2,c_3$, $E=\pi(P_i)$ and $C_0=P_i+R$.
\end{itemize}
Then the morphism $Z^{(k)}\to Y^{(10)}$, $E'\mapsto C_0+\pi^*E'$, where $k=\deg E$, induces a surjection of the completed local
rings at $\overline E$ and $\overline C$.
\end{lemma}

\begin{proof}
Surjectivity can be checked after the unramified base change to the ring of integers $W$ of the unramified extension of
$\Q_2$ of degree $6$, over which all the points involved split into points defined over $W$. Near $\overline E$ a divisor
$E'$ decomposes as a sum of divisors near the geometric points of $\overline E$, each of them given by a monic polynomial in a
local parameter $w$ of $Z$ whose degree is the multiplicity of the point in $\overline E$, and with non-leading coefficients
in the maximal ideal. It suffices to show that these polynomials can be recovered from the local equations of
$C_0+\pi^*E'$.

a) As $\pi$ is unramified at $c_1,c_2,c_3$ and at the geometric points of $q$, the pull-back of $w$ is a local parameter
$u$ at each of the points above. Some $a_i$ equals $m$, so $C_0$ does not contain $c_i$, and near $c_i$ the divisor
$C_0+\pi^*E'$ is given by $f(u)$, where $f(w)$ is the polynomial of the part of $E'$ near $\pi(c_1)$. Similarly $C_0$ does
not contain $q$, and near each geometric point of $q$ the divisor $C_0+\pi^*E'$ is given by the polynomial of the part of $E'$
near $\pi(q)$.

b) Near each of the $2$ geometric points of $P_i$, as $\tau$ has order prime to $2$ and multiplies the tangent space by a
primitive cube root of unity $\zeta$, there is a local parameter $z$ with $\tau^*z=\zeta z$, and $w=z^3$ is a local parameter of
$Z$. The part of $E'$ near the image of this point is given by $w-a$, and near the point the divisor $C_0+\pi^*E'$ is given by
$z(z^3-a)=z^4-az$, whose coefficient of $z$ recovers $a$.
\end{proof}

\begin{proposition}\label{prop:91}
The group $\torz{91}$ is not a subgroup of the torsion of an elliptic curve over a number field of degree $10$.
\end{proposition}

\begin{proof}
Let $D$ be the trace divisor of a point of order $91$ over a number field of degree $10$, let $D'+C'$ be its reduction modulo
$2$, and let $Y$, $\pi_X$, $\tau$ and $\pi$ be as in \Cref{lem:fi-91-Y}. The divisor $A=\pi_{X*}D$ is an effective rational
divisor of degree $10$ on $Y$ whose support contains no cusp, and its reduction is $\pi_{X*}(D'+C')$. By
\Cref{lem:fi-91-Y}~d), either $D'=0$, or $D'$ is a closed point of degree $8$ and $C'$ has degree $2$, or $D'$ is a closed point
of degree $9$ and $C'$ is a cusp of degree $1$.

Suppose first that $D'=0$. By \Cref{lem:fi-91-Y}~b), the reduction of $A$ is
$\overline C=a_1\overline{c_1}+a_2\overline{c_2}+a_3\overline{c_3}+b\overline q$ with $a_1+a_2+a_3+3b=10$, and it is the
reduction of the rational cuspidal divisor $C=a_1c_1+a_2c_2+a_3c_3+bq$. The class $(\tau-1)[C-10c_0]$ is cuspidal, so it is
torsion by the Manin-Drinfeld theorem. With $m$, $E$ and $C_0$ as in \Cref{lem:fi-91-immersion}~a) we have $k=\deg E=m+b$,
and $C_0$ is cuspidal of degree $10-3k$, which is positive, as $3\nmid10$. By \Cref{lem:fi-91-ranks}~a) and
\Cref{lem:fi-91-immersion}~a), \Cref{lem:fi-relative} applies and gives $A\geq C_0$, which is impossible, as the support of $A$
contains no cusp.

Suppose now that $D'$ is a closed point of degree $8$. The closed cusps of $X_1(91)_{\F_2}$ of degree at most $2$ have degree
$1$, so by \Cref{lem:fi-91-Y}~b) and~d) the reduction of $A$ is $4p_i+\overline R$ for some $i$ and an effective divisor
$\overline R$ of degree $2$ supported on $\overline{c_1},\overline{c_2},\overline{c_3}$. It is the reduction of $C=4P_i+R$, where
$R$ is the rational cuspidal divisor which reduces to $\overline R$. As $\tau P_i=P_i$, the class
$(\tau-1)[C-10c_0]=[\tau R-R]-10[\tau c_0-c_0]$ is torsion. With $E$ and $C_0$ as in \Cref{lem:fi-91-immersion}~b) we have
$k=2$, and \Cref{lem:fi-91-ranks}~b), \Cref{lem:fi-91-immersion}~b) and \Cref{lem:fi-relative} give $A\geq C_0\geq R$, which is
impossible.

Suppose finally that $D'$ is a closed point $P$ of degree $9$ and $C'$ is the reduction of a rational cusp $c$. Put
$\sigma=\dm3^{-1}$. The class $[\sigma D-D]=(\sigma-1)[D-10c_0]+10[\sigma c_0-c_0]$ is torsion, as $(\dm3-1)J_1(91)(\Q)$ is finite
(see the proof of \cite[Proposition 8.5]{N69}) and by the Manin-Drinfeld theorem, so it is annihilated by $A_3$, and as $A_3$ annihilates the rational cusps $c$ and $\sigma c$ as
divisors, the divisor $A_3(\sigma P-P)$ on $X_1(91)_{\F_2}$ is principal. In the proof of \cite[Proposition 8.5]{N69} it is
shown that this is not the case for any non-cuspidal closed point $P$ of degree $9$, which is a contradiction.
\end{proof}

%-------------------------------------------------------------------------------
\subsection{The curve \texorpdfstring{$X_1(93)$}{X1(93)}}\label{subsec:fi-93}

\begin{lemma}\label{lem:fi-93-cusps}
Let $X=X_1(93)$, let $h=\dm{32}$, which has order $2$ in $\Delta(93)$, and let
\[
U=(\Cart-1)\bigl((\dm2^*-1)H^0(X_{\F_2},\Omega^1)+(\dm5^*-1)H^0(X_{\F_2},\Omega^1)\bigr).
\]
\begin{itemize}
\item[a)] The closed cusps of $X_{\F_2}$ have degrees $1^{30},2^{15},5^6,10^3$, and $h$ maps every closed cusp of degree $2$ or
$10$ to itself. The automorphism $h$ fixes no non-cuspidal geometric point of $X$ in characteristic $0$.
\item[b)] We have $\dim U=248$, and the restriction map $U\to H^0(\overline C,\Omega^1|_{\overline C})$ is surjective for every
effective divisor $\overline C$ of degree $10$ on $X_{\F_2}$ supported on the closed cusps of degree $1$ and $5$.
\end{itemize}
\end{lemma}

\begin{proof}
a) The assertions on the cusps follow from their description, as we
\gitlink{continuation_20261030c/C93/contc93_cusp_descent.py}{check}. As $32^2\equiv1\pmod{93}$, we have
$32^2\not\equiv-1$ and $32^2\pm32+1\not\equiv0\pmod{93}$, so the last assertion follows from \cite[Lemma 7.2]{N69}.

b) We use the model given by $q$-expansions, as in \Cref{subsec:fi-85}, and we
\gitlink{continuation_20261030c/C93/contc93_cusp_blocks.py}{compute} that $\dim U=248$. The space $U$ is stable under the
diamond operators, so it suffices to prove the claim for one divisor in each diamond orbit. If $\overline C$ is supported
on the $30$ rational cusps, we may assume that $\infty$ has the largest multiplicity in it, and we compute the $155$ linear
forms $\lambda_{\infty,j}$ with $j\leq10$ and $\lambda_{c,j}$ with $j\leq5$ for the other $29$ rational cusps $c$ on $U$. They
span a space of dimension $150$, and with $w$ as in the proof of \Cref{lem:fi-85-cusps} we
\gitlink{continuation_20261030c/C93/contc93_cusp_FI.py}{check} that $w\geq30$ for each of the $2^5-1=31$ non-trivial
linear relations between them. As in the proof of \Cref{lem:fi-85-cusps}, this proves the claim for these divisors. The $6$ cusps of degree $5$ form one diamond orbit. If $\overline C$ contains one of them, we may assume that it
contains a fixed one, and then $\overline C$ is the sum of this cusp and of a divisor of degree $5$ on the rational cusps, or of
this cusp and one of the $6$ cusps of degree $5$. We compute the expansions of the differentials at the cusps of degree $5$ to
$2$ coefficients from the \gitlink{continuation_20261030c/C93/contc93_arc_q5.m}{Tate curve} $E(q^3)$ with the point
$\zeta_{31}q$, using the modular units of \cite[Sections 2 and 2.1]{DvH}, and we
\gitlink{continuation_20261030c/C93/contc93_cusp_FI.py}{check} that the restriction of $U$ is surjective at each of the
$\binom{34}{5}+6=278262$ divisors of these forms.
\end{proof}

\begin{proposition}\label{prop:93}
The group $\torz{93}$ is not a subgroup of the torsion of an elliptic curve over a number field of degree $10$.
\end{proposition}

\begin{proof}
Let $X=X_1(93)$, let $D$ be the trace divisor of a point $x$ of order $93$ over a number field of degree $10$, and let $D'+C'$
be its reduction modulo $2$. By \Cref{rem:exactdegree}, $D$ is a closed point of degree $10$. By the proof of
\cite[Proposition 8.5]{N69}, the only factor of $J_1(93)$ outside the winding quotient has trivial character, so
$(g-1)J_1(93)(\Q)$ is finite for every diamond operator $g$.

The non-cuspidal closed points of $X_{\F_2}$ of degree at most $9$ are $10$ points of degree $9$, which form one diamond orbit
\cite[Proposition 8.5]{N69}, and we \gitlink{computations/contc_parent_source_census.py}{compute} that there are none of
degree $10$. Suppose that $D'\neq0$. Then $D'$ is a closed point $P$ of degree $9$, and $C'$ is the reduction of a rational
cusp $c$. Put $\sigma=\dm2^{-1}$, $\tau=\dm5$ and $A_5=T_5-1-5\tau$. As in the proof of \Cref{prop:85}, the class
$[\sigma D-D]$ is torsion and annihilated by $A_5$, and the reduction of $A_5(\sigma D-D)$ is $A_5(\sigma P-P)$, so
$A_5(\sigma P-P)$ is principal. In the proof of \cite[Proposition 8.5]{N69} it is shown that this is not the case, which is
a contradiction.

So $D'=0$. Let $L=\Q(\zeta_{93})$, over which all the cusps of $X$ are defined, and let $\lambda$ be a prime of $L$ above $2$.
As the cuspidal subscheme is finite \'etale at $2$, the divisor $C'$ is the reduction modulo $\lambda$ of a unique cuspidal
divisor $C$ defined over $L$. Suppose first that $C'$ contains a closed cusp of degree $2$ or $10$, and let $h$ be as in
\Cref{lem:fi-93-cusps}. We follow the proof of \Cref{lem:stab}~c). The class
$y=(h-1)[D-C]=(h-1)[D-10c_0]-(h-1)[C-10c_0]\in J_1(93)(L)$ is torsion, by the finiteness of $(h-1)J_1(93)(\Q)$ and the
Manin-Drinfeld theorem, and it reduces to $0$ modulo $\lambda$, so $2y=0$ by the result of Parent recalled in
\Cref{subsec:hecke}, as $L$ is unramified above $2$. So $2hD+2C\sim2D+2hC$ over $L$. Let $G$ be the largest effective divisor with $G\leq C$ and $G\leq hC$. By
\Cref{lem:fi-93-cusps}~a), $h$ fixes the closed cusp of degree $2$ or $10$ contained in $C$, so $\deg G\geq2$. The divisors
$2hD+2(C-G)$ and $2D+2(hC-G)$ are effective, linearly equivalent and of degree at most $36$. As Abramovich's bound gives
$\gon_L X\geq\gon_\C X>38.09>36$, they are equal by \Cref{lem:closed}, and comparing their non-cuspidal parts we get $hD=D$.
So $h$ maps the closed point $D$ to itself, and it induces an automorphism $\eta$ of its residue field $K=\Q(x)$, of degree
$10$. As $h$ fixes no geometric point of $D$ by \Cref{lem:fi-93-cusps}~a), $\eta$ has order $2$, and its fixed field $K_0$ has
degree $5$. The morphism $f:X_1(93)\to X_1(31)$, $(E,P)\mapsto(E,3P)$, satisfies $f\circ h=f$, as $32\equiv1\pmod{31}$. So the
residue field of the non-cuspidal point $f(x)$ is contained in $K_0$, and there is an elliptic curve over a number field of
degree $1$ or $5$ with a point of order $31$. This is impossible, as $31\notin S(1)\cup S(5)$ \cite{mazur,DKSS}.

Suppose finally that $C'$ is supported on the closed cusps of degree $1$ and $5$. By the finiteness of $(\dm2-1)J_1(93)(\Q)$
and $(\dm5-1)J_1(93)(\Q)$ we may apply \Cref{prop:cartier} with $H=\{1\}$, $r=2$, $\psi_1=\dm2-1$ and $\psi_2=\dm5-1$, and by
\Cref{lem:fi-93-cusps}~b) it shows that $D$ does not reduce to $C'$, which is a contradiction.
\end{proof}