%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Second part of the section "The Hecke sieve and push-forwards to smaller levels":
% the groups Z/58, Z/60, Z/62, Z/66, Z/70 and Z/74.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{The curves \texorpdfstring{$X_1(58)$ and $X_1(62)$}{X1(58) and X1(62)}}\label{subsec:sw-58-62}

Over nonic fields the groups $\torz{58}$ and $\torz{62}$ are excluded in \cite[Proposition 6.11]{N69} by pushing
the trace divisor forward to $X_1(29)$, respectively $X_1(31)$, and comparing the classes of the possible reductions
modulo $3$ with the reduction of the Mordell-Weil group. We do the same in degree $10$. The new cases are the closed
points of degree $10$ and, for $X_1(62)$, the sums of two closed points of degree $5$. Some classes survive, and they
are the reductions of known points of $X_1(29)$ and $X_1(31)$.

\begin{lemma}\label{lem:sw-5862-setup}
Let $N\in\{29,31\}$, put $n=2N$ and $Y=X_1(N)$, and let $\pi:X_1(n)\to Y$ be the morphism $(E,P)\mapsto(E,2P)$. Let
$H\subseteq J_Y(\F_3)$ be the reduction of $J_Y(\Q)$.
\begin{enumerate}
\item[a)] We have $\gon_\Q X_1(29)=11$ and $\gon_\Q X_1(31)=12$. The groups
$$J_1(29)(\Q)\simeq\torz4\times\torz4\times\torz{64427244},\qquad J_1(31)(\Q)\simeq\torz{10}\times\torz{1772833370}$$
are generated by the classes of rational cuspidal divisors, and reduction modulo $3$ is injective on them. We have
$$
\begin{gathered}
J_1(29)(\F_3)\simeq(\torz4)^3\times(\torz8)^2\times\torz{386563464},\\
J_1(31)(\F_3)\simeq(\torz2)^2\times\torz{10}\times\torz{3352427902670},
\end{gathered}
$$
in which $H$ has index $2^9\cdot3$, respectively $2^2\cdot31\cdot61$.
\item[b)] The closed cusps of $Y_{\F_3}$ have degrees $1^{N'},N'$, where $N'=(N-1)/2$, and those of degree $1$ are the
reductions of the $N'$ rational cusps of $Y$.
\item[c)] Let $Q$ be a non-cuspidal closed point of $X_1(n)_{\F_3}$ of degree $e\leq10$, and let $P=\pi(Q)$, a closed
point of degree $k$. If $N=29$, then either $k=e\in\{6,\dots,10\}$ and the elliptic curve corresponding to $P$ has a
point of order $2$ over the residue field of $P$, or $k=3$, $e=9$ and $\pi_*Q=3P$. If $N=31$, then the elliptic curve
corresponding to $P$ has a point of order $2$ over the residue field of $P$, and either $k=e\in\{5,\dots,10\}$, or
$k=5$, $e=10$ and $\pi_*Q=2P$.
\end{enumerate}
\end{lemma}

\begin{proof}
The gonalities are \cite[Theorem 4 and Table 1]{DvH}, the Mordell-Weil groups and their generation by rational
cuspidal divisors are \cite[Theorem 3.1(2), Corollary 4.14 and Table 2]{DEHMZ}, and injectivity follows from Katz's
result, as $J_Y(\Q)$ is finite. The groups $J_Y(\F_3)$ and the indices of $H$ are computed in the proof of
\cite[Proposition 6.11]{N69}, with Magma's proven class group algorithm (see
\githubbare{continuation_20261031a/C58/inputs/C58_X129/build_presentation.m} and its analogue for $N=31$). Part b)
holds as $3$ has order $N-1$ modulo $N$ (see \cite[Proposition 6.11]{N69}).

For c), note that the fibre of $\pi$ above a non-cuspidal point $(E,P_N)$ consists of the points $(E,P_N'+T)$, where
$P_N'$ is the unique point of order $N$ with $2P_N'=P_N$ and $T$ runs over $E[2]\setminus\{0\}$. Hence $e=rk$, where $r\in\{1,2,3\}$
is the degree of the irreducible factor of the $2$-division polynomial of the elliptic curve corresponding to $P$ that
belongs to $Q$, and $\pi_*Q=rP$. By \cite[Table 3]{N69}, the curve $X_1(58)_{\F_3}$ has no non-cuspidal closed points of
degree at most $5$, the curve $X_1(62)_{\F_3}$ none of degree at most $4$, and $X_1(29)_{\F_3}$ and $X_1(31)_{\F_3}$ none of
degree at most $2$, respectively $3$. If $r=2$, then the $2$-division polynomial at $P$ is the product of a linear and
a quadratic factor, and the linear factor gives a non-cuspidal closed point of $X_1(n)_{\F_3}$ of degree $k=e/2\leq5$,
so $N=31$ and $k=5$. If $r=3$, then $k\leq 3$, so $N=29$ and $k=3$. In the remaining case $r=1$ and $k=e$. In all the cases
with $N=31$ the $2$-division polynomial at $P$ has a root, which proves c).
\end{proof}

\begin{lemma}\label{lem:sw-58-classes}
Let $Y=X_1(29)$, let $c_0$ be a rational cusp of $Y$, and let $H$ be as in \Cref{lem:sw-5862-setup}. Let $P$ be a
non-cuspidal closed point of $Y_{\F_3}$ of degree $k$, and assume that either $k\in\{6,\dots,10\}$ and the elliptic curve
corresponding to $P$ has a point of order $2$ over the residue field of $P$, or $k=3$. Then
$[P-k\,\overline{c_0}]\notin H$ if $k\geq6$, and $[3P-9\,\overline{c_0}]\notin H$ if $k=3$, except for $21$ closed points $P$ of
degree $10$. These $21$ points are the reductions of the closed points of degree $10$ of $X_1(29)$ in the
$\Delta(29)$-orbits of the points \xpoint{1.29.10.a} and \xpoint{1.29.10.b} of van Hoeij \cite{vanHoeij,x1mn}, and the
elliptic curves corresponding to these two points have no point of order $2$ over their residue fields.
\end{lemma}

\begin{proof}
For $k\leq9$ these are the $14$, $84$, $168$, $476$ and $1582$ tests in degrees $3$, $6$, $7$, $8$ and $9$ of
\cite[Proposition 6.11]{N69}, none of which lies in $H$ (see \githubbare{continuation_20261031a/C58/inputs/C58_X129/classify58.m}).
Note that $H$ is stable under the diamond operators and contains $[\dm a\overline{c_0}-\overline{c_0}]$, so
$[\dm aP-k\,\overline{c_0}]=\dm a[P-k\,\overline{c_0}]+k[\dm a\overline{c_0}-\overline{c_0}]$ lies in $H$ if and only if
$[P-k\,\overline{c_0}]$ does, and it suffices to test one point in each $\Delta(29)$-orbit. We
\gitlink{continuation_20261031a/C58/degree10_filter.m}{compute} that $Y_{\F_3}$ has $5971$ non-cuspidal closed points of
degree $10$, in $430$ orbits under $\Delta(29)$, that the elliptic curve has a point of order $2$ over the residue field
at $4060$ of them, which form $293$ orbits, and that for the representatives of exactly $2$ of these $293$ orbits, of
lengths $7$ and $14$, the class $[P-10\,\overline{c_0}]$ lies in $H$ (see \githubbare{jobs/d31a_C58_class_filter.log}).

The points \xpoint{1.29.10.a} and \xpoint{1.29.10.b} have residue fields $\Q(a)$ with
$$
\begin{aligned}
f_a&=a^{10}-2a^9+2a^8-5a^7+7a^6-4a^5+4a^4-8a^3+5a^2-1,\\
f_b&=a^{10}-5a^9+15a^8-31a^7+47a^6-56a^5+51a^4-37a^3+20a^2-7a+1.
\end{aligned}
$$
We \gitlink{continuation_20261031a/C58/known_lift_binding_job_v3.m}{compute} that $f_a$ and $f_b$ are irreducible modulo
$3$, and that the reductions of the $14$ diamond translates of these two points are the $7$, respectively $14$, points of
the two surviving orbits. We also find that $f_a$ and $f_b$ have the simple roots $11$, respectively $5$, modulo $29$, and at the
corresponding primes of degree $1$ the elliptic curves have good reduction with $29$ points. As reduction is injective
on the torsion of order prime to $29$, these curves have no point of order $2$ over their residue fields.
\end{proof}

\begin{proposition}\label{prop:58}
The group $\torz{58}$ is not a subgroup of the torsion of an elliptic curve over a number field of degree $10$.
\end{proposition}

\begin{proof}
Let $E/K$ have a point $P_{58}$ of order $58$, where $[K:\Q]=10$, let $D$ be the trace divisor of $x=(E,P_{58})$ on
$X_1(58)$, and let $Y$, $\pi$, $H$ be as in \Cref{lem:sw-5862-setup} with $N=29$. Then $D_Y:=\pi_*D$ is the trace divisor
of $\pi(x)=(E,2P_{58})$. By \Cref{lem:sw-5862-setup}~a) and \Cref{prop:main}~a), applied to $Y$ with $p=3$ and $d=10$,
the reduction of $D_Y$ modulo $3$ is not cuspidal. So the reduction of $D$ is $D'+C'$ with $D'\neq0$, and by
\Cref{lem:sw-5862-setup}~c) every non-cuspidal closed point of $X_1(58)_{\F_3}$ of degree at most $10$ has degree at least $6$, so
$D'$ is a single closed point $Q$ of degree $e\in\{6,\dots,10\}$ and $C'$ has degree $10-e\leq4$. The divisor $\pi_*C'$ is
cuspidal of degree at most $4$, so by \Cref{lem:sw-5862-setup}~b) it is supported on the reductions of the rational cusps,
and its class, normalized at a rational cusp $c_0$, lies in $H$. By \Cref{lem:classtest} applied to $D_Y$ we get
$[\pi_*Q-e\,\overline{c_0}]\in H$. By \Cref{lem:sw-5862-setup}~c) the divisor $\pi_*Q$ is a closed point $P$ of degree $e\geq6$ at
which the elliptic curve has a point of order $2$ over the residue field, or $3P$ with $P$ of degree $3$, so by
\Cref{lem:sw-58-classes}, $e=10$, $C'=0$, and $\pi_*Q=\overline B$ for a closed point $B$ of degree $10$ of $Y$ in the
$\Delta(29)$-orbit of one of the points \xpoint{1.29.10.a} and \xpoint{1.29.10.b}.

So $D_Y$ and $B$ have the same reduction, and as $[D_Y-B]\in J_Y(\Q)$ and reduction modulo $3$ is injective on $J_Y(\Q)$,
we get $D_Y\sim B$. As $\gon_\Q Y=11>10$, \Cref{lem:closed}~a) gives $D_Y=B$. Hence $\pi(x)$ is a $K$-rational point of
$Y$ above the closed point $B$, and as $Y$ is a fine moduli space, $E$ is isomorphic over $K$ to a conjugate of the
elliptic curve of $B$ over its residue field. By \Cref{lem:sw-58-classes} this curve has no point of order $2$ over its
residue field, which contradicts $29P_{58}\in E(K)$.
\end{proof}

\begin{lemma}\label{lem:sw-62-classes}
Let $Y=X_1(31)$, let $c_0$ be a rational cusp of $Y$, and let $H$ be as in \Cref{lem:sw-5862-setup}. Let $\mathcal P_k$
be the set of non-cuspidal closed points of $Y_{\F_3}$ of degree $k$ at which the elliptic curve has a point of order $2$ over
the residue field.
\begin{enumerate}
\item[a)] For $5\leq k\leq 10$ and $P\in\mathcal P_k$ we have $[P-k\,\overline{c_0}]\notin H$, except for $5$ points of
degree $9$. These $5$ points are the reductions of the $15$ diamond translates of the closed point $B_{31}$ of degree $9$
given by the point $x_{31}$ of \cite[Proposition 13.5]{N69}.
\item[b)] For $P,P'\in\mathcal P_5$, not necessarily distinct, we have $[P+P'-10\,\overline{c_0}]\notin H$.
\end{enumerate}
\end{lemma}

\begin{proof}
For $k\leq9$ there are $90$, $90$, $180$, $630$ and $1200$ tests in degrees $5,\dots,9$, and the computation in the proof
of \cite[Proposition 6.11]{N69} shows that exactly $5$ classes lie in $H$, all of degree $9$, and that these $5$ points
are the reductions of the diamond translates of $B_{31}$ (see \githubbare{continuation_20261031a/C62/inputs/C62_X131/classify62.m}
and \githubbare{continuation_20261031a/C62/inputs/C62_X131/known31/verify_known31.py}). We also
\gitlink{continuation_20261031a/C62/known31_global.m}{check} that the residue field of $B_{31}$ is inert at $3$. As in the
proof of \Cref{lem:sw-58-classes} it suffices to test one point in each orbit under $\Delta(31)$, and we
\gitlink{continuation_20261031a/C62/degree10_filter_resume_v2.m}{compute} that $Y_{\F_3}$ has $5685$ non-cuspidal closed
points of degree $10$, in $379$ orbits, that $3795$ of them, in $253$ orbits, lie in $\mathcal P_{10}$, and that none of the
$253$ representatives has class in $H$ (see \githubbare{jobs/d31a_C62_class_resume_v2.log}). This proves a).

The set $\mathcal P_5$ has $90$ elements, so there are $4095$ unordered pairs $\{P,P'\}$ with possibly $P=P'$, and we
\gitlink{continuation_20261031a/C62/small_support_arithmetic.py}{compute} that none of the classes $[P+P'-10\,\overline{c_0}]$
lies in $H$.
\end{proof}

\begin{proposition}\label{prop:62}
The group $\torz{62}$ is not a subgroup of the torsion of an elliptic curve over a number field of degree $10$.
\end{proposition}

\begin{proof}
Let $E/K$ have a point $P_{62}$ of order $62$, where $[K:\Q]=10$, let $D$ be the trace divisor of $x=(E,P_{62})$ on
$X_1(62)$, and let $Y$, $\pi$, $H$ be as in \Cref{lem:sw-5862-setup} with $N=31$, and $D_Y=\pi_*D$. As in the proof of
\Cref{prop:58}, \Cref{prop:main}~a), applied to $Y$ with $p=3$, $d=10$ and $\gon_\Q Y=12$, shows that the reduction of $D$ is
$D'+C'$ with $D'\neq0$. The non-cuspidal closed points of $X_1(62)_{\F_3}$ of degree at most $10$ have degree at least $5$ by
\Cref{lem:sw-5862-setup}~c), so $D'$ is either a single closed point $Q$ of degree $e\in\{5,\dots,10\}$, or a sum $Q+Q'$ of two
closed points of degree $5$, possibly equal. In both cases $C'$ has degree at most $5$, so $\pi_*C'$ is supported on the
reductions of the rational cusps by \Cref{lem:sw-5862-setup}~b), and \Cref{lem:classtest} applied to $D_Y$ gives
$[\pi_*D'-e\,\overline{c_0}]\in H$, where $e=\deg D'$.

If $D'=Q+Q'$, then $\pi_*D'=P+P'$ with $P,P'\in\mathcal P_5$ by \Cref{lem:sw-5862-setup}~c), which is impossible by
\Cref{lem:sw-62-classes}~b). If $D'=Q$ and $\pi_*Q=2P$ with $P$ of degree $5$, then again $P\in\mathcal P_5$, and this is
excluded by \Cref{lem:sw-62-classes}~b) with $P'=P$. In the remaining case $\pi_*Q=P\in\mathcal P_e$, so by
\Cref{lem:sw-62-classes}~a) we have $e=9$ and $P=\overline{B'}$ for a diamond translate $B'$ of $B_{31}$, and $C'$ is a
cusp of degree $1$, the reduction of a rational cusp $c$. So $D_Y$ and $B'+c$ have the same reduction, hence $D_Y\sim B'+c$
by the injectivity of reduction on $J_Y(\Q)$, and $D_Y=B'+c$ by \Cref{lem:closed}~a), as $\gon_\Q Y=12>10$. This is a
contradiction, as the support of $D_Y$ contains no cusp.
\end{proof}

\subsection{The curve \texorpdfstring{$X_1(60)$}{X1(60)}}\label{subsec:sw-60}

A point of order $60$ gives, after an isogeny of degree $2$, a point of $X_1(2,30)$ over the same field, and we use the
quotients of $X_1(2,30)$ from \cite[Section 11.3]{N69}, where $\tg{2}{30}$ is excluded in the degrees $6,\dots,9$. Over
number fields of degree $10$ the group $\tg{2}{30}$ occurs (\Cref{prop:2x28-2x30}), so we have to use the point of order
$60$ itself. It enters at two places: through the quintic points of $X_1(30)$ (\Cref{lem:sw-60-quintic}) and through a
halving condition at the closed points of $X_1(2,30)$ over finite fields (\Cref{lem:sw-60-mixed}).

The following lemma collects the set-up of \cite[Section 11.3]{N69}.

\begin{lemma}\label{lem:sw-60-setup}
Let $X=X_1(2,30)$, whose points are triples $(E,P,Q)$ with $P$ of order $2$ and $Q$ of order $30$ such that $P\neq15Q$.
Let $\alpha:X\to W$ be the quotient of $X$ by the automorphism $(E,P,Q)\mapsto(E,15Q,P+Q)$ of order $3$, and let
$f:W\to X_1(15)$ be the morphism through which $(E,P,Q)\mapsto(E,2Q)$ factors. Let $c_W$ be a rational cusp of $W$, and
let $H\subseteq J_W(\Q)$ be the subgroup generated by the classes $[C-(\deg C)c_W]$, where $C$ runs over the Galois orbits
of cusps of $W$. Let $Y=X_1(30)$, let $y_0$ be a rational cusp of $Y$, and let $\pi_1,\pi_2,\pi_3:X\to Y$ be the morphisms
$$\pi_1(E,P,Q)=(E,Q),\qquad \pi_2(E,P,Q)=(E,P+16Q),\qquad\pi_3(E,P,Q)=(E,P+Q).$$
Then $W$ has genus $9$, $f$ is the double cover given by a square root of the discriminant of the universal elliptic curve
over $X_1(15)$ in Tate normal form, every cusp of $X_1(15)$ is totally ramified in $f$, the Galois orbits of cusps of $W$
have degrees $1,1,4,2,4,1,2,1$, and the following hold.
\begin{enumerate}
\item[a)] The group $J_X(\Q)$ is finite and generated by the classes of rational cuspidal divisors, and for every
effective rational divisor $D$ of degree $d$ on $X$ we have $[\alpha_*D-d\,c_W]\in H$.
\item[b)] The group $J_W(\Q)$ is finite, reduction modulo $7$ is injective on it, and
$$J_W(\F_7)\simeq(\torz2)^6\times\torz{102}\times\torz{10608},\qquad H\simeq(\torz2)^5\times\torz8.$$
\item[c)] Let $y$ be a non-cuspidal closed point of $W_{\F_7}$ of degree $k$. Then either there are $3$ closed points of
$X_{\F_7}$ above $y$, each of degree $k$, or there is one closed point $z$ above $y$, of degree $3k$, with $\alpha_*z=3y$.
\item[d)] We have $J_Y(\Q)\simeq\torz4\times\torz{8160}$, generated by the classes of rational cuspidal divisors, and
$J_Y(\F_7)\simeq(\torz4)^3\times\torz{136}\times\torz{16320}$, in which the reduction $G$ of $J_Y(\Q)$ has index $4352$.
The images in $J_Y(\F_7)/G$ of the sets $\mathcal S_0,\dots,\mathcal S_6$ of \eqref{eq:pushclass}, for $\overline{y_0}$, have $1$,
$9$, $41$, $67$, $68$, $68$ and $68$ elements.
\end{enumerate}
\end{lemma}

\begin{proof}
All of this is in \cite[Section 11.3]{N69}, apart from the set $\mathcal S_6$. The finiteness and the generation in a) are
\cite[Theorem 3.1(3), Proposition 4.16 and Table 3]{DEHMZ}, and $J_Y(\Q)$ in d) is \cite[Theorem 3.1(2), Corollary 4.14 and
Table 2]{DEHMZ}. We \gitlink{continuation_20261031b/C60/d31b_C60_merge_three_maps.py}{compute} the images of
$\mathcal S_0,\dots,\mathcal S_6$.
\end{proof}

\begin{lemma}\label{lem:sw-60-series}
Let $W$, $f$, $c_W$ and $H$ be as in \Cref{lem:sw-60-setup}, let $\iota$ be the non-trivial automorphism of $W$ over
$X_1(15)$, and let $C_1,\dots,C_8$ be the Galois orbits of cusps of $W$. Every class in $H$ is of the form $[L-10\,c_W]$
for an effective rational cuspidal divisor $L=\sum_iv_iC_i$ of degree $10$ on $W$. For a fixed choice of such divisors
$L$, one for each of the $256$ classes, put
$$B=\sum_i\lfloor v_i/2\rfloor f(C_i),\qquad C=\sum_i(v_i\bmod 2)\,C_i,$$
so that $L=f^*B+C$. Then exactly one of the following holds.
\begin{enumerate}
\item[a)] ($168$ classes) $C\neq0$, and every effective rational divisor linearly equivalent to $L$ is of the form
$f^*B'+C$ for an effective rational divisor $B'$ on $X_1(15)$.
\item[b)] ($4$ classes) $C=0$, $\deg B=5$, and every effective rational divisor linearly equivalent to $L$ is of the form
$f^*B'$ for an effective rational divisor $B'$ of degree $5$ on $X_1(15)$.
\item[c)] ($84$ classes) Both eigenspaces of $\iota$ on $L(\overline L)$ over $\F_7$ are non-zero, and
$\ell(\overline L)=2$ for $60$ of these classes and $\ell(\overline L)=3$ for the other $24$.
\end{enumerate}
\end{lemma}

\begin{proof}
We \gitlink{continuation_20261031b/C60/d31b_C60_degree10_series.m}{compute} the effective rational cuspidal divisors
of degree $10$, which represent all $256$ classes of $H$, and for the chosen representatives $L$ we compute
$\ell(\overline L)$ on $W_{\F_7}$ and the dimensions of the two eigenspaces of $\iota$ on $L(\overline L)$. As every cusp of
$X_1(15)$ is totally ramified in $f$, the $\iota$-invariant functions in $L(\overline L)$ are the functions $h\circ f$ with
$h\in L(\overline B)$. We find that for $168$ classes the $(-1)$-eigenspace is $0$ and $\ell(\overline L)=\ell(\overline B)=k$
with $k=2$, $3$ and $4$ for $72$, $64$ and $32$ of them, so that $\deg C=10-2k>0$, that for $4$ classes the
$(-1)$-eigenspace is $0$, $C=0$ and $\ell(\overline L)=\ell(\overline B)=5$, and that for the remaining $84$ classes both
eigenspaces are non-zero, of dimensions $1$ and $1$ for $60$ classes, and $1$ and $2$, in some order, for $24$ classes.

In the first two cases $\deg B=k\geq2$, so $\ell(B)=k$ by the Riemann-Roch theorem on the curve $X_1(15)$ of genus $1$.
Hence $f^*L(B)\subseteq L(L)$ has dimension $k$ over $\Q$, and by semicontinuity
\cite[Theorem III.12.8]{hartshorne} $\ell(L)\leq\ell(\overline L)=k$, so $L(L)=f^*L(B)$. Every effective rational
divisor linearly equivalent to $L$ is $L+\ddiv(h\circ f)=f^*(B+\ddiv h)+C$ for some $h\in L(B)$, which proves a) and b).
\end{proof}

\begin{lemma}\label{lem:sw-60-quintic}
Let $F$ be a number field of degree $5$, let $E/F$ be an elliptic curve with a point of order $30$ over $F$, and let
$M=F(E[2])$. Assume that $[M:F]=2$. Then no elliptic curve over $M$ which is $2$-isogenous to $E$ over $M$ has a point of
order $60$ over $M$.
\end{lemma}

\begin{proof}
By \cite[Proposition 9.2]{N5}, the pair $(F,E)$ is isomorphic to one of the pairs $(K_{30},E_{30})$ and $(K_{30},E_{30}')$
of \cite[Proposition 9.1]{N5}, where $K_{30}$ is the quintic field defined by $t^5+t^4-3t^3+3t+1$, so we may assume that $E$
is one of these two curves. As $E[2]\subseteq E(M)$, the curves $2$-isogenous to $E$ over $M$ are the $3$ curves $E/\dm T$
with $T\in E[2]\setminus\{0\}$. For both curves $E$ we \gitlink{additions/C60/tors60.m}{compute} with Magma's function
\texttt{TorsionSubgroup} that each of these $3$ curves has torsion subgroup $\torz{30}$ over $M$ (see
\githubbare{additions/C60/tors60.log}).
\end{proof}

\begin{lemma}\label{lem:sw-60-mixed}
Let $X$, $\alpha$, $Y$, $\pi_1,\pi_2,\pi_3$, $y_0$ and $G$ be as in \Cref{lem:sw-60-setup}, and let $\mathcal S_j\subseteq
J_Y(\F_7)$ be the sets of \eqref{eq:pushclass} for $\overline{y_0}$. Let $L$ be one of the $84$ divisors of
\Cref{lem:sw-60-series}~c), and let $\overline D$ be an effective divisor of degree $10$ on $X_{\F_7}$ which is not entirely
cuspidal and satisfies $\alpha_*\overline D\sim\overline L$. Write
$\overline D=D'+C'$, with $D'$ non-cuspidal of degree $e$ and $C'$ cuspidal. Then one of the following holds.
\begin{enumerate}
\item[a)] $[\pi_{i*}D'-e\,\overline{y_0}]\notin G+\mathcal S_{10-e}$ for some $i\in\{1,2,3\}$.
\item[b)] $C'=0$, and $D'$ contains with multiplicity $1$ a closed point $(E_1,P,Q)$, with residue field $\kappa$, such that
there is no point $S\in(E_1/\dm P)(\kappa)$ with $2S=\phi(Q)$ and $\widehat\phi(S)=Q$, where $\phi:E_1\to E_1/\dm P$ is the
quotient map and $\widehat\phi$ its dual.
\end{enumerate}
\end{lemma}

\begin{proof}
As $\alpha_*\overline D$ is effective, it is one of the members of the linear system $|\overline L|$ over $\F_7$, and there are
$60\cdot8+24\cdot57=1848$ of them by \Cref{lem:sw-60-series}~c). For each member $M$ we run through the effective divisors
$D'$ on $X_{\F_7}$ with non-cuspidal support such that $\alpha_*D'$ is the non-cuspidal part of $M$, using
\Cref{lem:sw-60-setup}~c): above a point $y$ of multiplicity $m$ in $M$, if there are $3$ points of the same degree above
$y$, then $m$ is distributed in all possible ways among them, and if there is one point $z$ above $y$, then $3\mid m$ and
$z$ has multiplicity $m/3$. The cuspidal part is not restricted. We
\gitlink{continuation_20261031b/C60/d31b_C60_mixed_fibres_v2.m}{find} $3414$ possible non-zero $D'$, supported on $1674$ closed
points of $X_{\F_7}$. For these we \gitlink{continuation_20261031b/C60/d31b_C60_three_maps_common_v2.m}{compute} the classes
$[\pi_{i*}D'-e\,\overline{y_0}]$, and \gitlink{continuation_20261031b/C60/d31b_C60_merge_three_maps.py}{find} that all three
lie in $G+\mathcal S_{10-e}$ for only $24$ of them. These $24$ divisors have degree $10$, so $C'=0$, and they are sums of
distinct closed points with multiplicity $1$, of degrees $2+2+3+3$ ($12$ of them) or $2+4+4$ ($12$ of them). For each of the
$72$ closed points occurring in them we \gitlink{continuation_20261031b/C60/d31b_C60_final_halving.m}{compute} the points $S$ as
in b), and find that each of the $24$ divisors contains a point for which there is no such $S$.
\end{proof}

\begin{proposition}\label{prop:60}
The group $\torz{60}$ is not a subgroup of the torsion of an elliptic curve over a number field of degree $10$.
\end{proposition}

\begin{proof}
Let $X$, $\alpha$, $W$, $f$, $Y$, $\pi_1,\pi_2,\pi_3$, $y_0$, $G$ and $H$ be as in \Cref{lem:sw-60-setup}, and let
$\mathcal S_j\subseteq J_Y(\F_7)$ be the sets of \eqref{eq:pushclass} for $\overline{y_0}$. Let $E_{60}/K$ have a point $R$ of
order $60$, where $[K:\Q]=10$. Let $\phi_0:E_{60}\to E=E_{60}/\dm{30R}$, let $P$ be the
non-zero point of $\ker\widehat{\phi_0}=\phi_0(E_{60}[2])$, and put $Q=\phi_0(R)$, a point of order $30$. Then
$\widehat{\phi_0}(15Q)=30R\neq0=\widehat{\phi_0}(P)$, so $P\neq15Q$, and $x=(E,P,Q)$ is a point of $Y_1(2,30)(K)$. As
$\ker\widehat{\phi_0}=\dm P$, there is an isomorphism $\theta:E/\dm P\to E_{60}$ over $K$ with $\theta\circ\phi=\widehat{\phi_0}$,
where $\phi:E\to E/\dm P$ is the quotient map, and then $\phi_0=\widehat\phi\circ\theta^{-1}$. Hence
\begin{equation}\label{eq:sw-60-half}
S_0:=\theta^{-1}(R)\in(E/\dm P)(K)\quad\text{satisfies}\quad 2S_0=\phi(Q)\quad\text{and}\quad\widehat\phi(S_0)=Q.
\end{equation}
By \cite{mazur}, \cite{kamienny,KM} and \cite[Theorem 1.1]{N5}, the group $\tg{2}{30}$ is not a subgroup of the torsion of an
elliptic curve over a number field of degree $1$, $2$ or $5$, so $\Q(x)=K$ and the trace divisor $D$ of $x$ on $X$ is a
closed point of degree $10$. As $\alpha$ is a cyclic cover of degree $3$, also $D_W:=\alpha_*D$ is a closed point of degree
$10$, and by
\Cref{lem:sw-60-setup}~a), b) and \Cref{lem:sw-60-series} it is linearly equivalent to one of the $256$ divisors $L$.

In case a) of \Cref{lem:sw-60-series}, $D_W\geq C>0$, which is impossible as the support of $D_W$ contains no cusp. In case
b), $D_W=f^*B'$ with $B'$ effective of degree $5$, and as $D_W$ is a closed point of degree $10$, $B'$ is a closed point of
degree $5$, with residue field $F\subseteq K$, $[K:F]=2$. As $X_1(15)$ is a fine moduli space, there is an elliptic curve
$E'/F$ with a point $R_{15}$ of order $15$ such that $(E',R_{15})\simeq(E,2Q)$ over $K$. The full $2$-torsion of $E$ is
defined over $K$, so the $2$-division polynomial of $E'$ splits over $K$, and as $[K:F]=2$, it has a root in $F$. So $E'$ has
a point $T$ of order $2$ over $F$, and $T+R_{15}$ has order $30$. By \cite[Theorem 1.1]{N5}, $\tg{2}{30}\not\subseteq E'(F)$,
so $F(E'[2])=K$. Now $E_{60}\simeq E/\dm P$ is $2$-isogenous over $K$ to $E'_K\simeq E$ and has the point $R$ of order $60$ over
$K$, which contradicts \Cref{lem:sw-60-quintic}.

In case c), let $\overline D$ be the reduction of $D$ modulo $7$. As $J_1(60)(\Q)$ is finite \cite[Theorem 3.1(2)]{DEHMZ} and
$\gon_\Q X_1(60)\geq\gon_\C X_1(60)\geq\tfrac{325}{2^{15}}\cdot1152>11.4$ by Abramovich's bound, \Cref{prop:main}~a), applied
to $X_1(60)$ with $p=7$, shows that $E_{60}$ has good reduction at some prime of $K$ above $7$, and so does the isogenous curve
$E$. So $\overline D$ is not entirely cuspidal. It satisfies $\alpha_*\overline D=\overline{D_W}\sim\overline L$, and by
\Cref{lem:classtest}, applied to $\pi_{i*}D$ as in \eqref{eq:pushclass}, its non-cuspidal part $D'$ satisfies
$[\pi_{i*}D'-e\,\overline{y_0}]\in G+\mathcal S_{10-e}$ for $i=1,2,3$. By \Cref{lem:sw-60-mixed}, $D'=\overline D$ contains
with multiplicity $1$ a closed point $\overline x$ for which there is no point $S$ as in \Cref{lem:sw-60-mixed}~b). As
$\overline x$ has multiplicity $1$ in $\overline D$, it is the reduction of $x$ modulo a prime $\mathfrak p$ of $K$ above $7$
which is unramified and whose residue field is the residue field $\kappa$ of $\overline x$ (\Cref{subsec:trace}). The
elliptic curve $E$ has good reduction at
$\mathfrak p$, and as $X$ is a fine moduli space, the reduction of $(E,P,Q)$ modulo $\mathfrak p$ is isomorphic over $\kappa$ to the
triple of $\overline x$. The reduction of $S_0$ of \eqref{eq:sw-60-half} modulo $\mathfrak p$ is then a point $S$ as in
\Cref{lem:sw-60-mixed}~b), which is a contradiction.
\end{proof}

\subsection{The curve \texorpdfstring{$X_1(66)$}{X1(66)}}\label{subsec:sw-66}

Over number fields of degree $7$, $8$ and $9$ the group $\torz{66}$ is excluded in \cite[Proposition 13.7]{N69}, which
determines all the points of these degrees on $X_1(33)$. In degree $10$ the curve $X_1(33)$, which has gonality $10$, has
infinitely many points of degree $10$, and the reduction of the trace divisor modulo $2$ can lie in a pencil of degree $10$. We use the
isogenous curve given by the point of order $2$, as in the proof of \cite[Proposition 13.17]{N69}, and the reduction modulo
$5$ of these pencils. As a field of degree $10$ can have automorphisms of even order, we need the following lemma, which we
also use for $\torz{74}$.

\begin{lemma}\label{lem:sw-twoisog}
Let $K$ be a number field, let $\mathfrak p$ be a prime of $K$ above $2$, let $E/K$ be an elliptic curve with good
reduction at $\mathfrak p$, and let $\phi:E\to E'$ be an isogeny of degree $2$ defined over $K$.
\begin{enumerate}
\item[a)] If $E$ has supersingular reduction at $\mathfrak p$, then $\mathfrak p$ is ramified over $2$.
\item[b)] Suppose that $E$ has ordinary reduction at $\mathfrak p$ and that $\End_{\overline K}(E)=\Z$, and let $\sigma$ be an
automorphism of $K$ with $\sigma(\mathfrak p)=\mathfrak p$. Then $E'$ is not isomorphic to $E^\sigma$ over $K$.
\end{enumerate}
\end{lemma}

\begin{proof}
Let $v$ be the valuation of $K$ at $\mathfrak p$, and put $e=v(2)$. The curve $E'$ also has good reduction at $\mathfrak p$. For an
isogeny $\psi:E_1\to E_2$ between elliptic curves over $K$ with good reduction at $\mathfrak p$ and invariant differentials
$\omega_1,\omega_2$ of their smooth models over the local ring at $\mathfrak p$, we have $\psi^*\omega_2=u_\psi\omega_1$ with
$u_\psi$ integral at $\mathfrak p$, as $\psi$ extends to the smooth models, and $v(u_\psi)$ does not depend on the choice of the
$\omega_i$. The reduction of $u_\psi$ is the multiplier of the reduction $\widetilde\psi$, so $v(u_\psi)=0$ if and only if
$\widetilde\psi$ is separable \cite[Proposition II.4.2(c)]{silverman}. As $\widehat\phi\circ\phi=[2]$, we have
$v(u_\phi)+v(u_{\widehat\phi})=v(2)=e$.

a) If $\widetilde E$ is supersingular, then $\widetilde E[2](\overline{\F}_2)=0$, so $[2]$ is purely inseparable on
$\widetilde E$, both $\widetilde\phi$ and $\widetilde{\widehat\phi}$ are inseparable, and $e=v(u_\phi)+v(u_{\widehat\phi})\geq2$.

b) If $\widetilde E$ is ordinary, then $\widetilde E[2](\overline{\F}_2)$ has order $2$, so $[2]$ has inseparable degree $2$
on $\widetilde E$. As inseparable degrees are multiplicative, exactly one of $\widetilde\phi$ and $\widetilde{\widehat\phi}$ is
separable, so $v(u_\phi)\in\{0,e\}$. Suppose that $E'\simeq E^\sigma$ over
$K$, and let $\psi:E\to E^\sigma$ be the composition of $\phi$ with such an isomorphism, so that $v(u_\psi)=v(u_\phi)$, as an
isomorphism has a multiplier which is a unit at $\mathfrak p$. Let $h$
be the order of $\sigma$. The composition
$$\psi^{\sigma^{h-1}}\circ\dots\circ\psi^{\sigma}\circ\psi:E\to E^{\sigma^h}=E$$
is an endomorphism of $E$ of degree $2^h$, so it is multiplication by an integer $m$ with $m^2=2^h$, and $h$ is even and
$m=\pm2^{h/2}$. Choose an invariant differential $\omega$ of the smooth model of $E$ over the local ring at $\mathfrak p$. As
$\sigma(\mathfrak p)=\mathfrak p$, the differential $\sigma^i(\omega)$ is one for $E^{\sigma^i}$, and with these choices the
multiplier of $\psi^{\sigma^i}$ is $\sigma^i(u_\psi)$, whose valuation is $v(u_\psi)$. Hence the multiplier $m$ of the
composition has valuation $h\,v(u_\psi)\in\{0,he\}$, while $v(m)=he/2$, which is a contradiction as $he>0$.
\end{proof}

\begin{lemma}\label{lem:sw-66-F2}
Let $X=X_1(33)$, let $c_0$ be a rational cusp of $X$, and let $H\subseteq J_X(\F_2)$ be the reduction of $J_X(\Q)$. Let
$C_1,\dots,C_{10}$ be the rational cuspidal divisors of degree $9$ of the proof of \cite[Proposition 13.7]{N69}, which are
$\Z$-linear combinations of Galois orbits of cusps, not effective. We call the closed points of degree $7$, $8$ and $9$ of
$X$ in the $\Delta(33)$-orbits of the points of level $33$ of \cite[Propositions 13.2, 13.3 and 13.5]{N69} the
\textit{known} points of degree $7$, $8$ and $9$. There are $10$, $5$ and $10$ of them, and the following hold.
\begin{enumerate}
\item[a)] We have $\gon_\Q X=10$, the group $J_X(\Q)\simeq\torz5\times\torz{42373650}$ is generated by the classes of rational
cuspidal divisors, and $J_X(\F_2)\simeq\torz{15}\times\torz{42373650}$, in which $H\simeq\torz5\times\torz{42373650}$, so
reduction modulo $2$ is injective on $J_X(\Q)$.
\item[b)] The Galois orbits of cusps over $\Q$ have degrees $1^{10},2^5,10^2$, and the closed cusps of $X_{\F_2}$ have degrees
$1^{10},2^5,5^2,10$. The non-cuspidal closed points of $X_{\F_2}$ of degrees $1,\dots,10$ number
$$0,\ 0,\ 0,\ 0,\ 2,\ 20,\ 30,\ 25,\ 50,\ 123,$$
and the elliptic curves corresponding to the $2$ points of degree $5$ are supersingular.
\item[c)] The effective divisors $\overline B$ of degree $10$ on $X_{\F_2}$ with $[\overline B-10\,\overline{c_0}]\in H$ whose
support contains a non-cuspidal point are the following $3224$ divisors, and each of them is the only effective divisor in its
linear equivalence class.
\begin{enumerate}
\item[i)] $\overline S+C'$, where $S$ is a known point of degree $7$, $8$ or $9$ and $C'$ is an effective cuspidal divisor of
degree $3$, $2$ or $1$, respectively, which is the reduction of an effective rational cuspidal divisor ($3100$ divisors).
\item[ii)] $P_i+\overline c$, where $c$ is a rational cusp and $P_1,\dots,P_{10}$ are closed points of degree $9$ with
$[P_i-9\,\overline{c_0}]=[\overline{C_i}-9\,\overline{c_0}]$ ($100$ divisors).
\item[iii)] $21$ closed points of degree $10$.
\item[iv)] $3$ divisors which contain a non-cuspidal closed point of degree $5$ with multiplicity $1$.
\end{enumerate}
\item[d)] Every effective cuspidal divisor $\overline B$ of degree $10$ on $X_{\F_2}$ with $[\overline B-10\,\overline{c_0}]\in H$ is
the reduction of an effective rational cuspidal divisor, and it is the only effective divisor in its linear equivalence
class, except for $10$ classes, each of which contains exactly $3$ effective divisors.
\end{enumerate}
\end{lemma}

\begin{proof}
For a) see \cite[Theorem 4 and Table 1]{DvH}, \cite[Theorem 3.1(2), Corollary 4.14 and Table 2]{DEHMZ} and the proof of
\cite[Proposition 13.7]{N69}, where $J_X(\F_2)$ and $H$ are computed. The Galois orbits of cusps and the closed cusps are as in the proof of
\cite[Proposition 13.7]{N69}, and we \gitlink{computations/d31b_C66_F2_frame_v2.m}{compute} the non-cuspidal closed points of
degree at most $10$. An elliptic curve over $\F_{32}$ with a point of order $33$ has $33$ points by the Hasse bound, so its
trace of Frobenius is $0$ and it is supersingular.

As every non-cuspidal closed point of $X_{\F_2}$ of degree at most $10$ has degree at least $5$, the non-cuspidal part of an
effective divisor of degree $10$ on $X_{\F_2}$ is $0$, a single closed point of degree $5,\dots,10$ (of degree $5$ possibly with
multiplicity $2$), or a sum of two distinct points of degree $5$. We \gitlink{computations/d31b_C66_F2_degree10_classes.py}{run}
through all the effective divisors of degree $10$ on $X_{\F_2}$, with all the cuspidal parts, and determine those with class
in $H$. There are $321230$ of them, in $321210$ classes. As an effective divisor $\overline B$ with $\ell(\overline B)=r$ lies in a
linear equivalence class with exactly $2^r-1$ effective divisors over $\F_2$, this shows that $\ell(\overline B)=1$ for all of
them except for the $30$ divisors in $10$ classes with $3$ effective divisors each, for which $\ell(\overline B)=2$, and these
$30$ divisors are cuspidal. Among the $321230$ divisors, $318006$ are cuspidal, and they are exactly the reductions of the
effective rational cuspidal divisors of degree $10$ with class in $H$, which proves d).

The other $3224$ divisors are $2700$, $300$, $200$ and $21$ divisors whose non-cuspidal part is a closed point of degree $7$,
$8$, $9$ and $10$, respectively, and the $3$ divisors of iv), namely $2$ divisors consisting of a closed point of degree $5$
and a closed cusp of degree $5$, and the sum of the $2$ non-cuspidal closed points of degree $5$. The points of degree $7$, $8$
and $9$ that occur are $10$, $5$ and $20$ closed points, and every effective cuspidal divisor of degree $3$, $2$ and $1$ occurs
with them. By the proof of \cite[Proposition 13.7]{N69}, these points are the reductions of the known points, together with
$10$ closed points $P_1,\dots,P_{10}$ of degree $9$ with $[P_i-9\,\overline{c_0}]=[\overline{C_i}-9\,\overline{c_0}]$. The effective cuspidal divisors of
degree at most $3$ on $X_{\F_2}$ are reductions of effective rational cuspidal divisors by b). This proves c).
\end{proof}

\begin{lemma}\label{lem:sw-66-F5}
Let $X=X_1(33)$, and let $C_1,\dots,C_{10}$ be as in \Cref{lem:sw-66-F2}. For every $i\in\{1,\dots,10\}$ and every
rational cusp $c$ of $X$ we have $\ell(\overline{C_i+c})=0$ on $X_{\F_5}$. Hence no effective rational divisor on $X$ is
linearly equivalent to $C_i+c$.
\end{lemma}

\begin{proof}
We \gitlink{computations/d31b_C66_F5_extra_RR.m}{compute} the $100$ Riemann-Roch spaces over $\F_5$. The last statement
follows by semicontinuity \cite[Theorem III.12.8]{hartshorne}.
\end{proof}

For the $10$ classes of \Cref{lem:sw-66-F2}~d) we use the reduction modulo $5$ and the morphisms $\rho_1$ and $\rho_2$
of the following lemma.

\begin{lemma}\label{lem:sw-66-pencils}
Let $X=X_1(33)$, and let $c_0$ be as in \Cref{lem:sw-66-F2}. Let $\rho_1,\rho_2:X_1(66)\to X$ be the morphisms
$$\rho_1(E,R)=(E,2R),\qquad \rho_2(E,R)=(E/\dm{33R},\ 2R\bmod\dm{33R}),$$
which are defined over $\Q$, map cusps to cusps and non-cusps to non-cusps, and commute with the diamond operators. There
are effective rational cuspidal divisors $L_A$ and $L_B$ of degree $10$ on $X$ with the following properties.
\begin{enumerate}
\item[a)] The classes $[\dm aL_A-10\,c_0]$ and $[\dm aL_B-10\,c_0]$, for $a\in(\torz{33})^\times$, reduce to the $10$
classes of \Cref{lem:sw-66-F2}~d), $5$ of them for $L_A$ and $5$ for $L_B$.
\item[b)] On $X_{\F_5}$ we have $\ell(\overline{L_A})=\ell(\overline{L_B})=2$, and there are functions $g_A\in L(\overline{L_A})$
and $g_B\in L(\overline{L_B})$ of degree $10$.
\item[c)] The fibres of $g_A$ above $0$, $1$, $\infty$ and of $g_B$ above $0$, $4$, $\infty$ are entirely cuspidal, and the
fibres of $g_A$ above $2$, $3$, $4$ and of $g_B$ above $1$, $2$, $3$ contain no cusp. The latter consist of closed points of
degrees $4+6$, $10$, $10$ and $2+8$, $3+3+4$, $2+8$, respectively, each with multiplicity $1$.
\item[d)] Let $g,g'\in\{g_A,g_B\}$, $a\in(\torz{33})^\times$ and $t,t'\in\PP^1(\F_5)$, with $g^{-1}(t)$ not containing a cusp.
Then there is no effective divisor $Z$ of degree $10$ on $X_1(66)_{\F_5}$ with $\rho_{1*}Z=g^{-1}(t)$ and
$\rho_{2*}Z=(g'\circ\dm a)^{-1}(t')$.
\end{enumerate}
\end{lemma}

\begin{proof}
We take for $L_A$ and $L_B$ effective rational cuspidal divisors representing two of the $10$ classes, found in
\githubbare{computations/d31b_C66_prepare_gonal_cusps.py}. We \gitlink{computations/d31b_C66_two_gonal_orbits_filter.m}{compute}
$L(\overline{L_A})$ and $L(\overline{L_B})$ on $X_{\F_5}$, the functions $g_A$ and $g_B$, their diamond translates $g\circ\dm a$
and the fibres of $g_A$ and $g_B$ above $\PP^1(\F_5)$, which proves b) and c), and we
\gitlink{computations/d31b_C66_gonal_genuine_replay_v3.py}{check} that the polar divisors of the $20$ functions $g\circ\dm a$
represent the $10$ classes as stated in a).

For d), suppose that $Z$ is such a divisor. As $\rho_{1*}Z=g^{-1}(t)$ and every point $y$ of $g^{-1}(t)$ has
multiplicity $1$ by c), above every such $y$ there is exactly one point $z_y$ of $Z$, and it has multiplicity $1$ and the same
degree as $y$. As $\rho_{2*}Z=(g'\circ\dm a)^{-1}(t')$, every $z_y$ satisfies $(g'\circ\dm a)(\rho_2(z_y))=t'$. The closed points
of $X_1(66)_{\F_5}$ above a non-cuspidal point $y$ of $X_{\F_5}$ correspond to the irreducible factors of the $2$-division
polynomial of the elliptic curve of $y$, and above the $11$ non-cuspidal points of the fibres of c) there are $23$ of them. For
each of these points $z$ we \gitlink{computations/d31b_C66_two_gonal_orbits_filter.m}{compute} $\rho_2(z)$, using the isogeny
whose kernel is given by the corresponding factor, and the values at $\rho_2(z)$ of the $20$ functions $g'\circ\dm a$. We find
that for all $g,t,g',a,t'$ as in d), the sum of the degrees of the points $z$ above the points of $g^{-1}(t)$ with
$(g'\circ\dm a)(\rho_2(z))=t'$ is at most $4$. But for $Z$ as above this sum is at least $\sum_y\deg z_y=\deg Z=10$, which is a
contradiction.
\end{proof}

The following lemma describes all the effective rational divisors of degree $10$ on $X_1(33)$ whose support contains no
cusp, in terms of their reductions modulo $2$.

\begin{lemma}\label{lem:sw-66-divisors}
Let $X=X_1(33)$, and let $L_A$ and $L_B$ be as in \Cref{lem:sw-66-pencils}. Let $D$ be an effective rational divisor of
degree $10$ on $X$ whose support contains no cusp, and let $\overline D$ be its reduction modulo $2$. Then one of the
following holds.
\begin{enumerate}
\item[a)] $\overline D$ contains a non-cuspidal closed point of degree $5$ with multiplicity $1$.
\item[b)] $D$ and $\overline D$ are closed points of degree $10$, and $\ell(\overline D)=\ell(\dm2\overline D)=1$.
\item[c)] $\overline D$ is cuspidal, and $D\sim\dm aL$ for some $a\in(\torz{33})^\times$ and $L\in\{L_A,L_B\}$.
\end{enumerate}
\end{lemma}

\begin{proof}
Let $c_0$ and $H$ be as in \Cref{lem:sw-66-F2}. By \Cref{lem:classtest}, $[\overline D-10\,\overline{c_0}]\in H$, so
$\overline D$ is one of the divisors of \Cref{lem:sw-66-F2}~c) or d). Recall that reduction modulo $2$ is injective on
$J_X(\Q)$ by \Cref{lem:sw-66-F2}~a).

Suppose that $\overline D=\overline S+C'$ as in \Cref{lem:sw-66-F2}~c)~i), and let $C$ be an effective rational cuspidal
divisor with $\overline C=C'$. Then $[D-S-C]\in J_X(\Q)$ reduces to $0$, so $D\sim S+C$. By semicontinuity
$\ell(D)\leq\ell(\overline D)=1$, so $D=S+C$, which is impossible as $C\neq0$ is cuspidal. If $\overline D=P_i+\overline c$
as in c)~ii), then in the same way $D\sim C_i+c$, which is impossible by \Cref{lem:sw-66-F5}. If $\overline D$ is one of the
divisors of c)~iv), then a) holds.

Suppose that $\overline D$ is a closed point of degree $10$, as in c)~iii). The reduction of a closed point of $X$ of
degree $k$ is an effective divisor of degree $k$, and $\overline D$ is a single closed point with multiplicity $1$, so $D$ is a
closed point of degree $10$. We have $\ell(\overline D)=1$ by \Cref{lem:sw-66-F2}~c). The divisor $\dm2\overline D$ is also a non-cuspidal closed point of degree $10$, and its class
relative to $\overline{c_0}$ lies in $H$, as $H$ is stable under the diamond operators and contains
$[\dm2\overline{c_0}-\overline{c_0}]$. So $\ell(\dm2\overline D)=1$ by \Cref{lem:sw-66-F2}~c), and b) holds.

Suppose finally that $\overline D$ is cuspidal, and let $C$ be an effective rational cuspidal divisor with
$\overline C=\overline D$ (\Cref{lem:sw-66-F2}~d)). Then $D\sim C$, and if $\overline D$ is the only effective divisor in
its class, then $\ell(D)\leq1$ and $D=C$, which is impossible. So the class of $\overline D$ is one of the $10$ classes of
\Cref{lem:sw-66-F2}~d), and by \Cref{lem:sw-66-pencils}~a) and the injectivity of reduction, c) holds.
\end{proof}

\begin{proposition}\label{prop:66}
The group $\torz{66}$ is not a subgroup of the torsion of an elliptic curve over a number field of degree $10$.
\end{proposition}

\begin{proof}
Let $X=X_1(33)$, and let $\rho_1$, $\rho_2$ be as in \Cref{lem:sw-66-pencils}. Let $E/K$ have a point $R$ of order $66$,
where $[K:\Q]=10$. Put $P=2R$, of order $33$, let $\phi:E\to E_\phi=E/\dm{33R}$, which is defined over $K$, and
$P_\phi=\phi(P)$, of order $33$. Let $D$ and $D_\phi$ be the trace divisors of $(E,P)$ and $(E_\phi,P_\phi)$ on $X$, and
$D_{66}$ that of $(E,R)$ on $X_1(66)$, so that $\rho_{1*}D_{66}=D$ and $\rho_{2*}D_{66}=D_\phi$. As $E(K)$ has a point of
order $66$, \Cref{lem:CM} shows that $\End_{\overline K}(E)=\Z$. We denote reductions modulo $2$ by a bar, and apply
\Cref{lem:sw-66-divisors} to $D$.

Suppose that $\overline D$ contains a non-cuspidal closed point $y$ of degree $5$ with multiplicity $1$. By
\Cref{subsec:trace}, $y$ is the reduction of $(E,P)$ modulo a prime $\mathfrak p$ of $K$ above $2$ which is unramified, with
residue field $\F_{32}$. The curve $E$ has good reduction at $\mathfrak p$, and it is supersingular there by
\Cref{lem:sw-66-F2}~b). This contradicts \Cref{lem:sw-twoisog}~a).

Suppose that $D$ and $\overline D$ are closed points of degree $10$. By \Cref{subsec:trace}, $\overline D$ is the reduction of
$(E,P)$ modulo a prime $\mathfrak p$ of $K$ above $2$ which is unramified with residue field $\F_{2^{10}}$, so $\mathfrak p$
is the only prime of $K$ above $2$, and $E$ has good reduction at $\mathfrak p$. By \Cref{lem:sw-twoisog}~a) the reduction
is ordinary. We now argue as in the proof of \cite[Proposition 13.17]{N69}. The reduction of $\phi$ is an isogeny of
degree $2$ of an ordinary curve, so its kernel is either the connected subgroup of order $2$, and it is the Frobenius
isogeny, or the \'etale subgroup of order $2$, and it is the Verschiebung up to isomorphism. Hence the reduction of
$(E_\phi,P_\phi)$ is the image of the reduction of $(E,P)$ under the Frobenius, or under $\dm2$ composed with the inverse of
the Frobenius (see \cite[Section 2.1]{N69}). As closed points are stable under the Frobenius, $\overline{D_\phi}$ is
$\overline D$ or $\dm2\overline D$. As $\ell(\overline D)=\ell(\dm2\overline D)=1$ by \Cref{lem:sw-66-divisors}~b), the
injectivity of reduction on $J_X(\Q)$ (\Cref{lem:sw-66-F2}~a)) gives $D_\phi\sim D$, respectively $D_\phi\sim\dm2D$, and
semicontinuity gives $D_\phi=D$, respectively $D_\phi=\dm2D$. Hence the $K$-points $(E_\phi,P_\phi)$ and $(E,P)$,
respectively $(E,2P)$, of $X$ lie above the same closed point with residue field $K$, so they differ by an automorphism
$\sigma$ of $K$. As $X$ is a fine moduli space, $E_\phi\simeq E^\sigma$ over $K$, and $\sigma(\mathfrak p)=\mathfrak p$ as
$\mathfrak p$ is the only prime of $K$ above $2$. This contradicts \Cref{lem:sw-twoisog}~b).

Suppose finally that $\overline D$ is cuspidal, so that $D\sim\dm aL$ with $a$ and $L$ as in \Cref{lem:sw-66-divisors}~c).
The elliptic curve $E$ has bad reduction at every prime of $K$ above $2$, which is multiplicative by \Cref{lem:additive},
and so has the isogenous curve $E_\phi$. So $\overline{D_\phi}$ is cuspidal too, and \Cref{lem:sw-66-divisors}, applied to
$D_\phi$, gives $D_\phi\sim\dm{a'}L'$ for some $a'\in(\torz{33})^\times$ and $L'\in\{L_A,L_B\}$. Replacing $R$ by $bR$, where
$b$ is coprime to $66$ and $b\equiv a^{-1}\pmod{33}$, replaces $D$, $D_\phi$ and $D_{66}$ by their diamond translates, so we
may assume that $a=1$.

Now we reduce modulo $5$, and denote the reductions by $D^{(5)}$, $D_\phi^{(5)}$ and $D_{66}^{(5)}$. Let $g\in\{g_A,g_B\}$ be
the function of \Cref{lem:sw-66-pencils}~b) with $g\in L(\overline L)$, so that $\overline L$ is the polar divisor of $g$. As
$D\sim L$ and $\ell(\overline L)=2$, the divisor $D^{(5)}$ is linearly equivalent to $\overline L$ on $X_{\F_5}$, so it equals
$\overline L+\ddiv(\lambda+\mu g)$ for some $(\lambda,\mu)\neq(0,0)$ in $\F_5^2$, that is, $D^{(5)}=g^{-1}(t)$ for some
$t\in\PP^1(\F_5)$. In the same way $D_\phi^{(5)}=(g'\circ\dm{a'}^{-1})^{-1}(t')$ for some $t'\in\PP^1(\F_5)$, where $g'$ is the
function with polar divisor $\overline{L'}$. If $g^{-1}(t)$ is entirely cuspidal, then $E$ has bad reduction at all the primes
of $K$ above $5$, which contradicts \Cref{prop:main}~a) applied to $X_1(66)$ with $p=5$, as $J_1(66)(\Q)$ is finite
\cite[Theorem 3.1(2)]{DEHMZ} and $\gon_\Q X_1(66)\geq\tfrac{325}{2^{15}}\cdot1440>14.2$ by Abramovich's bound. Otherwise
$g^{-1}(t)$ contains no cusp by \Cref{lem:sw-66-pencils}~c), and $Z=D_{66}^{(5)}$ contradicts \Cref{lem:sw-66-pencils}~d).
\end{proof}

\subsection{The curve \texorpdfstring{$X_1(70)$}{X1(70)}}\label{subsec:sw-70}

For $X_1(70)$ we push the reduction of the trace divisor forward to $X_1(35)$ along two maps, and use the Hecke operator
$A_2$ there. The prime $q=2$ is not covered by \Cref{prop:tools}, but the following lemma suffices.

\begin{lemma}\label{lem:sw-70-A2}
The operator $A_2=T_2-2\dm2-1$ annihilates $J_1(35)(\Q)$, and $A_2C=0$ as a divisor for every rational cuspidal divisor $C$
on $X_1(35)$.
\end{lemma}

\begin{proof}
By \Cref{lem:cuspHecke} with $q=2$, we have $A_2C=(\sigma_2C-C)+2\dm2(\sigma_2^{-1}C-C)=0$ for every rational cuspidal
divisor $C$. The group $J_1(35)(\Q)$ is generated by the classes of rational cuspidal divisors of degree $0$
\cite[Theorem 3.1(2), Corollary 4.14 and Table 2]{DEHMZ}, so $A_2$ annihilates it.
\end{proof}

Let $\pi_1,\pi_2:X_1(70)\to X_1(35)$ be the morphisms
$$\pi_1(E,P)=(E,2P),\qquad\pi_2(E,P)=(E/\dm{35P},\ 2P\bmod\dm{35P}).$$
Note that $2P\bmod\dm{35P}$ has order $35$, as $\dm{35P}$ has order $2$. Both maps are defined over $\Q$, map cusps to cusps and
non-cusps to non-cusps, and commute with the diamond operators.

\begin{lemma}\label{lem:sw-70-test}
\begin{enumerate}
\item[a)] The Galois orbits of cusps of $X_1(70)$ over $\Q$ and the closed cusps of $X_1(70)_{\F_3}$ both have degrees
$1^{24},4^6,6^4,12^2$.
\item[b)] The non-cuspidal closed points of $X_1(70)_{\F_3}$ of degrees $1,\dots,10$ number
$$0,\ 0,\ 0,\ 0,\ 0,\ 48,\ 408,\ 735,\ 2144,\ 5880,$$
and those of degrees $6,\dots,10$ form $4$, $34$, $65$, $180$ and $490$ orbits under $\Delta(70)$.
\item[c)] For every non-cuspidal closed point $P$ of $X_1(70)_{\F_3}$ of degree between $6$ and $10$, one of the divisors
$A_2(\pi_{1*}P)$ and $A_2(\pi_{2*}P)$ on $X_1(35)_{\F_3}$ is not principal.
\end{enumerate}
\end{lemma}

\begin{proof}
Part a) is a direct \gitlink{independent_checks/cusp_degrees.py}{computation} with the description of the cusps of
$X_1(70)$ in \cite[Section 2]{DN}: the Galois orbits
over $\Q$ are the orbits of $\Gal(\Q(\zeta_{70})/\Q)$ on the geometric cusps, and the closed cusps of $X_1(70)_{\F_3}$ are the
orbits of $\sigma_3$. We \gitlink{computations/d31a_even_p3_census_common.m}{compute} b). As $A_2$ and the $\pi_i$ commute with the diamond operators,
it suffices to prove c) for one point in each of the $773$ orbits. For each of them we
\gitlink{computations/d31a_C70_A2_common.m}{compute} the divisor $A_2(\pi_{1*}P)$ of degree $0$ directly from the moduli
interpretation of $T_2$, using the three subgroups of order $2$ of the elliptic curve, and find that $\ell(A_2(\pi_{1*}P))=0$ in
$770$ cases. In the remaining $3$ cases, all of degree $8$, we find that $\ell(A_2(\pi_{2*}P))=0$.
\end{proof}

\begin{proposition}\label{prop:70}
The group $\torz{70}$ 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 $Y_1(70)$ over a number field of degree $10$. We have $\rk J_1(70)(\Q)=0$
\cite[Theorem 3.1(2)]{DEHMZ}, and Abramovich's bound gives $\gon_\Q X_1(70)\geq\tfrac{325}{2^{15}}\cdot1728>17.1$, so by
\Cref{prop:main}~a) with $p=3$ the reduction of $D$ modulo $3$ is $D'+C'$ with $D'\neq0$ non-cuspidal. By
\Cref{lem:sw-70-test}~b), $D'$ is a single closed point of degree $6,\dots,10$ with multiplicity $1$, and $C'$ has degree at
most $4$. By \Cref{lem:sw-70-test}~a), $C'$ is the reduction of a rational cuspidal divisor $C$.

Let $i\in\{1,2\}$. The class $[\pi_{i*}D-10\,c_0]$, for a rational cusp $c_0$ of $X_1(35)$, lies in $J_1(35)(\Q)$, so by
\Cref{lem:sw-70-A2} the divisor $A_2(\pi_{i*}D)$ is principal, as $A_2c_0=0$. Its reduction modulo $3$ is
$A_2(\pi_{i*}D')+A_2(\pi_{i*}C')$, and $A_2(\pi_{i*}C')$ is the reduction of $A_2(\pi_{i*}C)=0$, as $\pi_{i*}C$ is a rational
cuspidal divisor. So $A_2(\pi_{i*}D')$ is principal for $i=1,2$, which contradicts \Cref{lem:sw-70-test}~c).
\end{proof}

\subsection{The curve \texorpdfstring{$X_1(74)$}{X1(74)}}\label{subsec:sw-74}

The Jacobian $J_1(37)$ has positive rank, so we work with the operator $\gamma:=\dm2-1$ on $J_1(37)$, which has finite image
on $J_1(37)(\Q)$, and with a formal immersion argument at $2$ for the map $B\mapsto\gamma[B-10\,c_0]$. The cases that it does
not cover are treated with the Hecke sieve at $2$, with \Cref{lem:stab}, and with the isogenous curve given by the point of
order $2$, as for $\torz{66}$. Some of the computations on $X_1(37)_{\F_2}$ that we use do not depend on the degree, and were
carried out together with computations for nonic fields.

\begin{lemma}\label{lem:sw-74-J}
Let $X=X_1(37)$, $J=J_1(37)$, let $c_0$ be a rational cusp of $X$, and put $\gamma=\dm2-1$.
\begin{enumerate}
\item[a)] We have $\gon_\Q X=18$, and $(\dm a-1)J(\Q)$ is finite for every $a\in(\torz{37})^\times$.
\item[b)] We have $J(\Q)[2]=0$, and reduction modulo $2$ is injective on $J(\Q)_\tors$.
\item[c)] The Galois orbits of cusps over $\Q$ and the closed cusps of $X_{\F_2}$ both have degrees $1^{18},18$. The
non-cuspidal closed points of $X_{\F_2}$ of degrees $1,\dots,10$ number
$$0,\ 0,\ 0,\ 0,\ 0,\ 21,\ 18,\ 0,\ 39,\ 99.$$
Under $\Delta(37)$ those of degree $6$ form $2$ orbits, of lengths $18$ and $3$, those of degree $7$ form $1$ orbit, of
length $18$, and those of degree $9$ form $4$ orbits, of lengths $18$, $18$, $2$ and $1$.
\item[d)] For the representatives $P$ of the orbits of length $18$ of the points of degrees $6$, $7$ and $9$, the divisor
$\gamma A_3P$ on $X_{\F_2}$ is not principal.
\item[e)] The diamond operator $\dm{10}$ has order $3$ in $\Delta(37)$, and it fixes every point in the orbits of lengths
$3$, $2$ and $1$ of c).
\end{enumerate}
\end{lemma}

\begin{proof}
The gonality is \cite[Theorem 1]{DvH}. By \cite[Proposition 5.2 and Table 2]{DSt} the prime $37$ is not strange, that is,
every newform of weight $2$ and level $37$ with non-trivial character has non-vanishing $L$-value at $1$. By Kato's theorem the
factors of $J$ on which the diamond operators act non-trivially have finite Mordell-Weil groups over $\Q$, so
$(\dm a-1)J(\Q)$ is finite. This proves a).

For b), $A_3$ and $A_5$ annihilate $J(\Q)_\tors$ by \Cref{prop:tools}~b), and we
\gitlink{continuation_20261031b/C74/inputs/work/degree9/prime_orders/continued_20261026/deck/rational_two_diamond.m}{compute}, using
the action of the Hecke algebra on $H_1(J(\C),\Z)/2H_1(J(\C),\Z)\simeq J[2]$, that the kernels of $A_3$ and $A_5$ on $J[2]$
intersect trivially (see \githubbare{continuation_20261031b/C74/inputs/work/degree9/prime_orders/continued_20261026/P37/rational_two_delta4.log}).
So $J(\Q)[2]=0$, and as the kernel of reduction modulo $2$ on $J(\Q)_\tors$ has exponent dividing $2$ by Parent's result, it is
trivial.

In c), the statement on the cusps holds as $2$ has order $36$ modulo $37$. An elliptic curve over $\F_{2^k}$ with a point of
order $37$ does not exist for $k\leq5$ and for $k=8$, by the Hasse bound and as the possible even traces of Frobenius over
$\F_{2^k}$ are $0$ and $\pm2^{(k+1)/2}$ for odd $k$ and $0$, $\pm2^{k/2}$ and $\pm2^{k/2+1}$ for even $k$
\cite[Theorem 4.1]{waterhouse}. We \gitlink{continuation_20261031b/C74/inputs/work/degree9/prime_orders/continued_20261026/prime_good_branches_64GB.m}{compute}
the points of degrees $6$, $7$ and $9$ and their orbits, \gitlink{additions/C74/stab37.m}{the lengths} of these orbits (see
\githubbare{additions/C74/stab37.log}), and \gitlink{computations/d31b_C74_formal_export_v4.m}{the points} of degree $10$.

For d) we \gitlink{continuation_20261031b/C74/inputs/work/degree9/prime_orders/continued_20261026/prime_good_branches_64GB.m}{compute}
$A_3P$ directly from the moduli interpretation of $T_3$, and the Riemann-Roch spaces of $\gamma A_3P$. The representatives
are numbered in the same way in the
\gitlink{continuation_20261031b/C74/inputs/work/degree9/prime_orders/continued_20261026/P37/good_branches_64GB.log}{log} of
this computation and in \githubbare{additions/C74/stab37.log}.

For e), as $10^3=1000\equiv1\pmod{37}$ and $10\not\equiv\pm1\pmod{37}$, the operator $\dm{10}$ has order $3$ in
$\Delta(37)\simeq(\torz{37})^\times/\{\pm1\}$, which is cyclic of order $18$. The stabilizer of a point in an orbit of length
$3$, $2$ or $1$ is the subgroup of $\Delta(37)$ of order $6$, $9$ or $18$, which contains the unique subgroup of order $3$,
generated by $\dm{10}$.
\end{proof}

The image $U$ of $\dm2^*-1$ on $H^0(X_1(37)_{\F_2},\Omega^1)$, where $\dm2^*$ is the pull-back of differentials by the
diamond automorphism $\dm2$, is the image of the action of $\gamma$ on the cotangent space at $0$ of the special fibre of the
N\'eron model of $J_1(37)$ over $\Z_2$, identified with $H^0(X_1(37)_{\F_2},\Omega^1)$, and it is stable under the diamond
operators. The following is the formal immersion criterion of \cite[Proposition 7.3]{N69}, in the simpler situation in which
reduction is injective on the torsion.

\begin{lemma}\label{lem:sw-74-fi}
Let $X=X_1(37)$, $J=J_1(37)$, $c_0$ and $\gamma=\dm2-1$ be as in \Cref{lem:sw-74-J}, and let $U\subseteq
H^0(X_{\F_2},\Omega^1)$ be the image of $\dm2^*-1$. Let $\overline B$ be an effective divisor of degree $10$ on $X_{\F_2}$
such that the restriction map $U\to H^0(\overline B,\Omega^1|_{\overline B})$ is surjective. Let $D_1$ and $D_2$ be
effective rational divisors of degree $10$ on $X$ which both reduce to $\overline B$ modulo $2$, and such that $\gamma[D_1-D_2]$ is
torsion. Then $D_1=D_2$.
\end{lemma}

\begin{proof}
The class $\gamma[D_1-D_2]\in J(\Q)_\tors$ reduces to $\gamma[\overline B-\overline B]=0$, so it is $0$ by
\Cref{lem:sw-74-J}~b). Consider the morphism $h:X^{(10)}\to J$ over $\Z_{(2)}$ given by $h(B)=\gamma[B-10\,c_0]$, where $X$ and $J$
denote the smooth models. Then $h(D_1)=h(D_2)$, and $D_1$ and $D_2$ have the same reduction $\overline B$. The cotangent map of
$B\mapsto[B-10\,c_0]$ at $\overline B$ is the restriction map $H^0(X_{\F_2},\Omega^1)\to H^0(\overline B,\Omega^1|_{\overline B})$,
so the image of the cotangent map of $h$ at $\overline B$ is the restriction of $U$ to $\overline B$. By assumption it is
surjective, so $h$ is a formal immersion at $\overline B$, and by \cite[Lemma 4.2]{DKSS} (which is due to Parent
\cite[Lemme 4.13]{parentformal}) $h$ is injective on the residue class of $\overline B$. Hence $D_1=D_2$.
\end{proof}

\begin{lemma}\label{lem:sw-74-ranks}
Let $X=X_1(37)$, and let $U\subseteq H^0(X_{\F_2},\Omega^1)$ be the image of $\dm2^*-1$. Then $\dim U=38$, and the
restriction map $U\to H^0(\overline B,\Omega^1|_{\overline B})$ is surjective for the following effective divisors $\overline B$
of degree $10$ on $X_{\F_2}$:
\begin{enumerate}
\item[a)] every divisor supported on the $18$ cusps of degree $1$,
\item[b)] every non-cuspidal closed point of degree $10$.
\end{enumerate}
\end{lemma}

\begin{proof}
We \gitlink{computations/d31b_C74_formal_export_v4.m}{compute} a basis of $H^0(X_{\F_2},\Omega^1)$, which has dimension $40$, the
matrix of $\dm2^*$ on it, of order $18$, and its image $U$, of dimension $38$. As $U$ is stable under the diamond operators,
which act transitively on the cusps of degree $1$, it suffices to compute the expansions at one cusp of degree $1$, and to
transport them with the diamond operators. We compute the expansions of a basis of $U$ at a cusp of degree $1$ up to order
$10$, and at each of the $99$ points of degree $10$, and we \gitlink{computations/d31b_C74_prefix_rank_driver.py}{check} that the
restriction maps have rank $10$ for all the $\binom{27}{10}=8436285$ divisors of a) (with the program
\githubbare{computations/d31b_C74_prefix_ranks.cpp}), and for all the $99$ divisors of b).
\end{proof}

\begin{proposition}\label{prop:74}
The group $\torz{74}$ is not a subgroup of the torsion of an elliptic curve over a number field of degree $10$.
\end{proposition}

\begin{proof}
Let $X=X_1(37)$, $J=J_1(37)$, $c_0$ and $\gamma$ be as in \Cref{lem:sw-74-J}. Let $E/K$ have a point $R$ of order $74$, where
$[K:\Q]=10$. Put $P=2R$, of order $37$, let $\phi:E\to E_\phi=E/\dm{37R}$, which is defined over $K$, and $P_\phi=\phi(P)$.
Let $D$ and $D_\phi$ be the trace divisors of $(E,P)$ and $(E_\phi,P_\phi)$ on $X$. By \Cref{rem:exactdegree}, $D$ is a
closed point of degree $10$. As $E(K)$ has a point of order $74$, \Cref{lem:CM} shows that $E$ has no complex
multiplication. Note that for every effective rational divisor $B$ of degree $10$ on $X$ the class $\gamma[D-B]$ lies in
$\gamma J(\Q)$, which is finite by \Cref{lem:sw-74-J}~a). We consider the reduction $\overline D$ of $D$ modulo $2$.

Suppose first that $\overline D$ is cuspidal. Then it is supported on the cusps of degree $1$, so it is the reduction of a rational
cuspidal divisor $C$, and \Cref{lem:sw-74-fi} and \Cref{lem:sw-74-ranks}~a) give $D=C$, which is impossible. So $\overline D=D'+C'$ with
$D'\neq0$, and by \Cref{lem:sw-74-J}~c), $D'$ is a single closed point of degree $6$, $7$, $9$ or $10$ with multiplicity $1$, and
$C'$ is supported on the cusps of degree $1$. Let $C$ be the rational cuspidal divisor with $\overline C=C'$.

The class $\gamma[D-10\,c_0]$ is torsion, so $A_3\gamma[D-10\,c_0]=0$ by \Cref{prop:tools}~b). As $\gamma c_0$ and
$\gamma C$ are rational cuspidal divisors, $A_3\gamma c_0=A_3\gamma C=0$ as divisors by \Cref{prop:tools}~c), so
$\gamma A_3(D-C)$ is principal, and so is its reduction $\gamma A_3D'$. As $\gamma A_3$ commutes with the diamond operators,
\Cref{lem:sw-74-J}~d) shows that $D'$ lies in one of the orbits of lengths $3$, $2$ and $1$ of \Cref{lem:sw-74-J}~c), of degree
$6$, $9$ and $9$, or has degree $10$.

If $\deg D'\in\{6,9\}$, then $C$ has degree $10-\deg D'\leq4$, and $g=\dm{10}$ fixes the reduction $D'$ of $D-C$ by
\Cref{lem:sw-74-J}~e). The automorphism $g$ has odd order $3$, it acts trivially on $J(\Q)[2]=0$, $(g-1)J(\Q)$ is finite, and
$\gon_\Q X=18>10+\deg C$. So \Cref{lem:stab}~d) gives $gD=D$. As $E$ has no complex multiplication, \Cref{lem:orbits} shows that
$3$ divides $\deg D=10$, which is a contradiction.

Finally, suppose that $D'=\overline D$ is a closed point of degree $10$. By \Cref{subsec:trace}, $\overline D$ is the reduction
of $(E,P)$ modulo a prime $\mathfrak p$ of $K$ above $2$ which is unramified with residue field $\F_{2^{10}}$, so $\mathfrak p$ is
the only prime of $K$ above $2$, and $E$ has good reduction at $\mathfrak p$. By \Cref{lem:sw-twoisog}~a) the reduction is
ordinary. As in the proof of \Cref{prop:66}, the reduction of $D_\phi$ is $\overline D$ or $\dm2\overline D$, and both are
closed points of degree $10$. Applying \Cref{lem:sw-74-fi} and \Cref{lem:sw-74-ranks}~b) to $D_\phi$ and $D$, respectively
$\dm2D$, we get $D_\phi=D$, respectively $D_\phi=\dm2D$. As in the proof of \Cref{prop:66}, it follows that
$E_\phi\simeq E^\sigma$ over $K$ for an automorphism $\sigma$ of $K$ with $\sigma(\mathfrak p)=\mathfrak p$, which
contradicts \Cref{lem:sw-twoisog}~b).
\end{proof}