# A Comptonized Fireball Bubble: Physical Origin of Magnetar Giant Flares

Zhao Joseph Zhang (张钊) School of Astronomy and Space Science, Nanjing
University, Nanjing 210023, China  Key Laboratory of Modern Astronomy and
Astrophysics (Nanjing University), Ministry of Education, China  Bin-Bin Zhang
School of Astronomy and Space Science, Nanjing University, Nanjing 210023,
China  Key Laboratory of Modern Astronomy and Astrophysics (Nanjing
University), Ministry of Education, China  Yan-Zhi Meng School of Astronomy
and Space Science, Nanjing University, Nanjing 210023, China  Key Laboratory
of Modern Astronomy and Astrophysics (Nanjing University), Ministry of
Education, China

###### Abstract

Magnetar Giant Flares (MGFs) have been long proposed to contribute at least a
sub-sample of the observed short Gamma-ray Bursts (GRBs). The recent discovery
of the short GRB 200415A in the nearby galaxy NGC 253 established a textbook-
version connection between these two phenomena. Unlike previous observations
of the Galactic MGFs, the unsaturated instrument spectra of GRB 200415A
provide for the first time an opportunity to test the theoretical models with
the observed $\gamma$-ray photons. This paper proposed a new readily fit-able
model for the MGFs, which invokes an expanding fireball Comptonized by the
relativistic magnetar wind at photosphere radius. In this model, a large
amount of energy is released from the magnetar crust due to the magnetic
reconnection or the starquakes of the star surface and is injected into
confined field lines, forming a trapped fireball bubble. After breaking
through the shackles and expanding to the photospheric radius, the thermal
photons of the fireball are eventually Comptonized by the relativistic
$e^{\pm}$ pairs in the magnetar wind region, which produces additional higher-
energy gamma-ray emission. The model predicts a modified thermal-like spectrum
characterized by a low-energy component in the Rayleigh-Jeans regime, a smooth
component affected by coherent Compton scattering in the intermediate energy
range, and a high-energy tail due to the inverse Compton process. By
performing a Monte-Carlo fit to the observational spectra of GRB 200415A, we
found that the observation of the burst is entirely consistent with our model
predictions.

Gamma-ray bursts; Gamma-ray transient sources; Magnetars, Giant flares

##  1 Introduction

Magnetars (Mereghetti et al., 2015; Kaspi & Beloborodov, 2017) belong to a
particular type of neutron stars carrying ultra-strong magnetic field
typically in the range of B $\sim 10^{14}$-$10^{15}$ G (Duncan & Thompson,
1992), much higher than the critical magnetic field, $B_{Q}=4.4\times 10^{13}$
G, above which the nonrelativistic Landau energy becomes comparable to the
electron rest energy (Thompson & Duncan, 1995). Observationally, magnetars are
registered as two classes: the Anomalous X-ray Pulsars (AXPs) and the Soft
Gamma-ray Repeaters (SGRs). SGRs have recently aroused increasing research
interest due to their associations with Fast Radio Bursts (FRBs; Israel et
al., 2016; Metzger et al., 2019; Katz, 2020; Lin et al., 2020; Zhang, 2020)
and Gamma-ray Bursts (GRBs; Hurley et al., 1999; Yang et al., 2020; Roberts et
al., 2021; Svinkin et al., 2021). In particular, the magnetar giant flares
(hereafter, MGFs) from the SGRs (Mazets et al., 1979a, b) have been long
considered to be a subclass of short GRBs (Laros et al., 1986; Atteia et al.,
1987).

The first magnetar giant flare (GRB 790305B) was observed from SGR 0526-66
(Golenetskii et al., 1984), which is located in the star-forming Dorado region
in the Large Magellanic Cloud (LMC). Since then, two more Galactic MGF events
have been confirmed, namely, the MGF of SGR 1900+14 on 1998 August 27 (Feroci
et al., 2002) and the MGF of SGR 1806-20 on 2004 December 27 (Yamazaki et al.,
2006). Located within a few $\sim$ kpc, those events, although having provided
enriched data in studying their temporal features (including the short abrupt
rise, the quasi-exponential decay, the subsequent pulsating tail; Hurley et
al. 1999; and the quasi-periodic oscillations; Israel et al. 2005; Strohmayer
& Watts 2005) of the MGFs, are all observed saturated by the $\gamma$-ray
detectors due to their overwhelmingly large numbers of photons. Thus, studying
the accurate spectral data of MGF has been infeasible until the extragalactic
MGF-originated event, GRB 200415A, was observed.

GRB 200415A is an apparent short gamma-ray burst discovered in the nearby
Sculptor galaxy (NGC 253), which is located about 3.5 Mpc away (Bissaldi et
al., 2020), much further than those in the previous MGF sample. A
comprehensive analysis performed by Yang et al. (2020) suggests the burst is a
significant outlier of both the Type I and Type II GRBs, but otherwise
entirely consistent with being an MGF event in terms of the temporal and
spectral features. Thanks to its significant distance outside the Milky Way,
the bright GRB 200415A is not subject to instrumental saturation and provides
an ideal case to study the photon behaviors of the MGFs. A spectral fit using
some empirical functions suggested that the burst is dominated by thermal-like
emissions (Yang et al., 2020), likely originates from an expanding fireball
outside the magnetar surface and may be Doppler-boosted by the relativistic
wind (Roberts et al., 2021).

Nevertheless, as a rare and most powerful type of SGR activity, MGFs’ physical
origin, trigger mechanism, and radiation process are still not fully unveiled.
A trigger of an MGF can be caused either internally (e.g., large-scale star
crust fracturing caused by the shear forces against the motion of the magnetic
footpoints; neutral point reconnection due to the torsional of the twisted
interior magnetic field; Parker, 1983a, b; Thompson & Duncan, 1995) or
externally (e.g., interchange instability and/or magnetic reconnection;
Moffatt, 1985; Thompson & Duncan, 2001). On the other hand, the radiation
process models of MGFs are relatively less diverse. Thompson & Duncan (1995)
proposed a “trapped fireball” model to explain the March 5th event on SGR
0526-66. The model introduces an initially expanding fireball to explain the
first sharp spike of the MGF event. The fireball is eventually trapped by
optically thick pair plasma in the stellar magnetosphere, producing repeated
pulsations. In addition, Thompson & Duncan (1995) suggested that a
relativistic wind driven by the pressure of the electron-position pair
($e^{\pm}$ hereafter) plasma from the NS surface, which can also release a
fraction of the star’s energy. The prediction of the Ep-flux correlation of
such a model was claimed to be consistent with the observation of GRB 200415A
(Roberts et al., 2021). Although these studies may shed light on the origin
and radiation mechanisms of the MGFs, a detailed first-principle calculation
of such a scenario and its validation through a direct fit to the observed
spectra is still missing to date.

Motivated by the previous studies, in this Letter, we studied in detail how
the fireball expands from the NS surface and penetrates the area of the
magnetar wind. Our finding suggests that the fireball is eventually
Comptonized by the dense $e^{\pm}$ pair plasma at a much larger radius and
produces a multi-component thermal-like radiation spectrum. Moreover, we
directly compare this model with the spectral data of GRB 200415A. This Letter
is organized as follows. We describe the physical picture of our model in §2.
In §3, we formulized the radiation mechanism for the calculation of the
specific flux and fit our model to the spectral data, constraining some
physical parameters. A brief summary is presented in §4.

##  2 The physical picture

Our model requires a magnetar characterized by a large-scale dipolar magnetic
field with some small-scale and axisymmetric magnetic topology (Gourgouliatos
et al., 2016). The small-scale field can be caused by internal motions such as
the hall drift of the crustal magnetic field. As illustrated in Figure 1, the
local small-scale magnetic field lines will be strongly wound up so that the
toroidal component is greater than the poloidal dipole strength. Therefore,
the local magnetic field may be an order of magnitude higher than that of the
large-scale dipolar field, which can exceed $10^{16}G$. Under such an
assumption, the physical picture of our model can be outlined in the following
steps:

Figure 1: An illustration of the magnetic topology of the magnetar in our
model.

  1. 1.

The occurrence of the small-scale magnetic instability. The Hall drift of the
interior field lines and the activity of the NS crust can make the local
external field become extremely unstable, causing strong magnetic
reconnection.

  2. 2.

The formation of the trapped fireball. Near the NS surface, an enormous amount
of energy carried by photons is instantly released via the small-scale
magnetic reconnection and interchange instability, and numerous $e^{\pm}$
pairs escaped from the NS due to the fracturing of the crust. These photons
and pairs, coupling with each other, are injected into the magnetosphere. A
fraction of the energy of the photon-rich pair plasma is confined by the
closed magnetic field lines and forms a trapped fireball as shown in figure 2.

Within the trapped fireball, the energy is gradually dissipated via thermal
radiation with a timescale of up to hundreds of seconds. A pulsating tail is
expected within such a long timescale due to the continuous local quasi-
periodic activities of the magnetar. The total energy carried by the pulsating
tail is roughly estimated as $E_{\rm tail}\sim 10^{44}$ ergs in the previous
three MGFs (Mereghetti, 2008). To confine this amount of energy in the closed
field lines, the pressure at the outer boundary of the field loop is required
to satisfy(Yang & Zhang, 2015)

$\frac{B^{2}_{R_{s}+l_{0}}}{8\pi}\gtrsim\frac{E_{\mathrm{tail}}}{3l_{0}^{3}},$
(1)

where $R_{s}$ is the surface radius of the magnetar, and $l_{0}$ is the scale
of the trapped fireball which in this case is smaller than $R_{s}$, i .e.,
$l_{0}<R_{s}$ (c.f., Boggs et al., 2007) . $B_{r}$ is defined by
$B_{r}=B_{*}(r/R_{s})^{-3}$ for a dipolar field at radius of $r$, where
$B_{*}$ is the characteristic surface magnetic field of the neutron star.

Observationally, the size of the fireball can be estimated by

$\begin{split}l_{0}\sim\Big{(}\frac{L_{\mathrm{\rm tail}}}{2\pi
caT_{\mathrm{tail}}^{4}}\Big{)}^{1/2}\sim&10^{5}\ {\rm
cm}\bigg{(}\frac{L_{\mathrm{tail}}}{5\times 10^{43}\mathrm{erg\
s^{-1}}}\bigg{)}^{1/2}\\\ &\times\bigg{(}\frac{kT_{\mathrm{tail}}}{70\
\mathrm{keV}}\bigg{)}^{-2},\end{split}$ (2)

where $a$ is the radiation constant, and $T_{\rm tail}$ is the thermal
equilibrium temperature which can be estimated by the cutoff energy
$T_{\mathrm{cut}}$. The characteristic values of $L_{\mathrm{tail}}\sim
5\times 10^{43}{\rm erg}\ {\rm s}^{-1}$ and $kT_{\rm tail}=70$ keV in Eq. (2)
are comparable to those observed in SGR 1806-20.

Figure 2: A schematic diagram of our model. A large number of high-energy
photons accompanied by electron-positron pairs are released near the NS
surface. A fraction of them is trapped in the confined magnetic field lines
(dark red region). The other part breaks through the magnetic field lines due
to sufficient radiation pressure and spreads out in the form of a relativistic
expanding fireball along the magnetic tube. A large amount of dense $e^{\pm}$
pairs (marked red and blue circle with $\pm$ signs) are emitted from the gap
region (red square), and accelerated by the gap potential difference. The
pairs further propagate along the open field lines and form a wind zone at a
larger radius (open blue region). The expanding fireball eventually penetrates
the wind area, and the photons are scattered by the massive pairs in the
emission region at photosphere radius (purple square), producing the high-
energy $\gamma$-ray photons.

  3. 3.

The formation of the magnetar wind. Due to the significant pressure of the
photon-pair plasma, numerous $e^{\pm}$ is driven from the magnetosphere. These
pairs, after being accelerated by the gap potential difference, produce the
high-energy curvature radiation, which is influenced by the magnetic field, in
turn, can be converted into secondary $e^{\pm}$ pairs. As a result, the
$e^{\pm}$ pairs increase rapidly in number and move along the field lines,
forming a relativistic wind (Figure 2. The magnetar wind will supply a high
number density of charged particles (i.e., electrons and positrons) which can
be calculated (Kumar & Bošnjak, 2020) as a function of the magnetic field,
$B_{*}$, the rotation period, $P$, and the distance to the center of the
magnetar, $R$, as

$\begin{split}&n_{\pm}=\frac{\mathcal{M}\bm{B_{0}}\cdot\Omega_{*}}{2\pi qc}\\\
&\approx 2\times 10^{25}\
\mathrm{cm}^{-3}\bigg{(}\frac{\mathcal{M}}{10^{9}}\bigg{)}\bigg{(}\frac{P}{1\
s}\bigg{)}^{-1}\bigg{(}\frac{B_{\mathrm{*}}}{5\times 10^{16}\
\mathrm{G}}\bigg{)}\\\
&\quad\times\bigg{(}\frac{R}{R_{\mathrm{s}}}\bigg{)}^{-3},\end{split}$ (3)

where $\mathcal{M}$ is the multiplicity parameter defined as the ratio of
maximum Lorentz factor of primary electons and secondary $e^{\pm}$ pairs,
which can be estimated up to $\sim 10^{9}$ for a magnetar (Ruderman &
Sutherland, 1975).

  4. 4.

The expanding of the fireball. The fireball will break through the shackles of
the field lines and expand along the magnetic tube when the radiation pressure
of the fireball is stronger than the magnetic pressure. The local magnetic
field lines become open during the expansion (figure 2). For a dipolar field,
the bulk Lorentz factor and comoving temperature are determined by the local
acceleration as well as the size of expanding fireball, $l$:

$\Gamma=(l/l_{0})^{3/2},\quad T=T_{\rm ini}(l/l_{0})^{-3/2},\\\ $ (4)

where $T_{\rm ini}$ and $l_{0}$ are the initial temperature and radius of the
trapped fireball, respectively. We set $kT_{\rm ini}=844$ keV, which is
obtained from observations of the GRB 200415A (Yang et al., 2020).

  5. 5.

The interaction between the expanding fireball and the wind. The fireball
expands relativistically and penetrates to the wind zone within the
magnetosphere. The photosphere radiation is generally considered to be
thermal; therefore, blackbody radiation is often used to describe the
spectrum. However, since an extraordinarily high density of $e^{\pm}$ pairs
with a thermal distribution is supplied by the magnetar wind, an extra finite
thermal medium (Compton cloud) emerges in the front of the expanding path of
the fireball, which is highly opaque and leads to a series of radiation
transfer effects including scatterings and absorption (Rybicki & Lightman,
1986; Beloborodov, 2010; Roberts et al., 2021). The occurrence of these energy
transfer effects will change the spectral shape significantly. The photons,
from either the thermal radiation of the fireball or the annihilation of
$e^{\pm}$ in the wind, are continuously scattered by a large number of
$e^{\pm}$ and finally escape at the photosphere in the form of thermal-like
radiation through coherent scattering (CC) and inverse Compton (IC)
scattering.

With a strong magnetic field and high number density of the pair plasma,
photons with electric vector perpendicular to the magnetic field (E-mode) are
easier to escape. The corresponding Rosseland mean optical depth (Duncan &
Thompson, 1992; Meszaros & Rees, 1992)) of the fireball with a size of $l$ is
expressed as

$\tau_{\perp}=\frac{4\pi^{2}}{5}\sigma_{\rm
T}\bigg{(}\frac{kT}{m_{e}c^{2}}\frac{B_{Q}}{B_{R}}\bigg{)}^{2}n_{\pm}l,\\\ $
(5)

where $\sigma_{\rm T}$ is the Thompson cross section, $m_{e}$ is the electron
mass, c is the light speed, and $B_{R}$ is the magnetic field at radius R,
i.e., $B_{R}=B_{*}\big{(}R/R_{s}\big{)}^{-3}$. By requiring $\tau_{\perp}=1$,
one can drive the minimal radiation radius of the expanding fireball as $l\sim
9\times 10^{5}$ cm, which is about one order of magnitude larger than $l_{0}$.

##  3 Radiation and Fit

Our model above predicts that the thermal photons of expanding fireball are
subject to the coherent Compton scattering (CC) and incoherent Inverse Compton
(IC) scattering, so the resulted spectrum is likely a modified blackbody
(hereafter, MB111We use “MB” to distinguish from multi-color blackbody (“MBB”;
Meng et al., 2018, 2019, 2021; Wang et al., 2021).) shape with an IC bump at
the high-energy end. A parameterized function of the model-predicted spectrum
can be derived by considering the above two radiation processes.

Whether the CC or IC dominates the radiation process are determined by the
Compton $y$ parameter, which varies at different photon energies. Following
Rybicki & Lightman (1986), we define

$y\equiv\frac{kT\overline{N}}{m_{e}c^{2}},$ (6)

where $k$ is the Boltzmann constant, and $\overline{N}$ is the mean scattering
number when a photon travels a free path.

Depending on the $y$ parameter, the derived flux can be considered in the
following two regimes:

  * •

For $y<$1, CC dominates, and the energy of a scattered photon is not
significantly changed. In this case, an MB spectral shape is expected. The
specific intensity of CC is closely related to the scattering and absorption
of the thermal photons (Rybicki & Lightman, 1986):

$I^{\rm CC}_{\nu}=\frac{2B_{\nu}}{1+\sqrt{(\kappa_{\rm ff}+\kappa_{\rm
es})\kappa_{\rm ff}^{-1}}},$ (7)

where $\kappa_{\rm ff}$ is the absorption factor of the free-free
(bremsstrahlung) process in a thermal medium, $\kappa_{\rm es}$ is the
scattering opacity from $e^{\pm}$ plasma, and $B_{\nu}$ is the Planck function
expressed as

$B_{\nu}=\frac{2h\nu^{3}/c^{2}}{\mathrm{exp}(h\nu/kT)-1},$ (8)

where $h$ is the Planck constant. $\kappa_{\rm ff}$ and $\kappa_{\rm es}$ can
be further written as

$\displaystyle\kappa_{\rm es}=\frac{\sigma_{\rm T}}{m_{e}}=730\
\mathrm{cm^{2}\ g^{-1}},$ (9)

and

$\displaystyle\kappa_{\rm ff}$
$\displaystyle=\frac{4q_{e}^{6}}{3{m_{e}}hc}\bigg{(}\frac{2\pi}{3{km_{e}}}\bigg{)}^{1/2}T^{-1/2}n_{\pm}^{2}\nu^{-3}$
$\displaystyle\quad\times(1-e^{-x})\overline{g}_{ff}(x)$
$\displaystyle=3.7\times
10^{8}T^{-1/2}n_{\pm}^{2}\nu^{-3}(1-e^{-x})\overline{g}_{ff}(x),$ (10)

where $q_{e}$ is the electron charge and $x\equiv h\nu/kT$. The $g_{\rm
ff}(x)$ is the free-free Gaunt factor, and for the energy range in this study,
it can be approximated as

$g_{\rm ff}(x)\sim 3\pi^{-1/2}\mathrm{ln}(2.25/x).$ (11)

Considering the Doppler boosting, the observed specific flux at a luminosity
distance, $D_{\rm L}$, can be calculated by

$\displaystyle F^{\rm CC}_{\nu_{\rm obs}}$ $\displaystyle=\frac{4\pi
D^{3}}{c^{2}}\frac{h(\nu_{\rm
obs}/D)^{3}}{\mathrm{exp}\bigg{(}\frac{h\nu_{obs}}{DkT}\bigg{)}-1}\Bigg{(}1+\sqrt{\frac{\kappa_{\rm
ff}+\kappa_{\rm es}}{\kappa_{\rm ff}}}\Bigg{)}^{-1}$
$\displaystyle\quad\times\bigg{(}\frac{l_{x}}{D_{\rm L}}\bigg{)}^{2},$ (12)

where $D$ is the Doppler factor. $F^{\rm CC}_{\nu_{\rm obs}}$ strongly depends
on the competition between $\kappa_{\rm ff}$ and $\kappa_{\rm es}$, which can
be easily noticed by introducing a characteristic frequency, $\nu_{\rm 0,CC}$,
at which the scattering and absorption coefficients are equivalent, i.e.,

$\kappa_{\rm es}=\kappa_{\rm ff}(\nu_{\rm 0,CC}).$ (13)

For $\nu<\nu_{\rm 0,CC}$, absorption is more dominant, and Eq. (7) approaches
the Rayleigh-Jeans limit and the observed spectrum (Eq. 12) is a pure
blackbody. On the other hand, for $\nu>\nu_{\rm 0,CC}$, the scattering
gradually becomes significant (see Figure 3) at higher energies and modifies
the spectrum (Eq. 12) significantly.

For a pair plasma with a typical density of $n_{\pm}=2.0\times 10^{25}$ cm-3
and temperature in range of 1-60 keV, it is calculated that $h\nu_{\rm 0,CC}$
is typically $\gtrsim$1.3 keV in the comoving frame, as shown in Figure 3.
Such a frequency falls well into the MeV range in the observer frame,
confirming that the CC process is indeed necessary to be taken into account.
This can also be verified by checking $x_{\rm 0,CC}\equiv h\nu_{\rm
0,CC}/kT\ll 1$, suggesting that photon at such a characteristic frequency is
significantly subject to scattering in a hot plasma.

Figure 3: Absorption coefficient, $\kappa_{\rm ff}$, as a function of photon
energy at different equilibrium temperatures. The $\kappa_{\rm ff}$ evolves
with the temperature, marked by the solid lines of different colors. The
dashed line represents the $\kappa_{\rm es}$. The dotted-dashed line marks the
characteristic frequency $\nu_{\rm 0,CC}$ at k$T=1$ keV.

  * •

For $y$ $\gg$ 1, the radiation is dominated by IC, and the energy is
transferred from $e^{\pm}$ pairs to the photons. In this case, the specific
intensity can be approximated by the Wien law (Rybicki & Lightman, 1986):

$I^{\rm IC}_{\nu}=\frac{2h\nu^{3}}{c^{2}}e^{-\alpha}e^{-x},$ (14)

where the factor $e^{-\alpha}$ is related to the local rate at which photons
are produced, defined by (Landau & Lifshitz, 1969)

$e^{-\alpha}=\frac{N}{V}\bigg{(}\frac{h^{2}}{2\pi mkT}\bigg{)}^{3/2}\sim
n_{c}\bigg{(}\frac{h^{2}}{2\pi m_{c}kT}\bigg{)}^{3/2}.$ (15)

where m is the particle mass, N is the total number of particles, and V is the
volume of the radiation region. $n_{c}$ and $m_{c}$ are defined as coupling
number density and coupling mass, respectively. $e^{-\alpha}$ measures
degeneracy of the particles. For example, $e^{-\alpha}$ is negligible if the
particles meet the classical limit condition and are entirely non-degenerate.
Otherwise, $e^{-\alpha}$ can be a significant number if the photons are
coupling with the $e^{\pm}$ pairs in a degenerate plasma.

The specific flux at the observer is

$F^{\rm IC}_{\nu_{\rm obs}}=\frac{8\pi h\nu_{\rm
obs}^{3}}{{c^{2}}}e^{-\alpha}\mathrm{exp}\bigg{(}\frac{-h\nu_{\rm
obs}}{DkT}\bigg{)}\bigg{(}\frac{l_{x}}{D_{\mathrm{L}}}\bigg{)}^{2}.$ (16)

Another critical frequency, $\nu_{\rm 0,IC}$, is defined by requiring
$y(\nu_{\rm 0,IC})=1$, above which IC process becomes significant. For pairs
which follow the nonrelativistic222In the comoving frame, the mean Lorentz
factor of the $e^{\pm}$ pairs, $\left\langle\gamma_{\pm}\right\rangle$, can be
estimated by $\left\langle\gamma_{\pm}\right\rangle=\sqrt{(12(kT/m_{\rm
e}c^{2})^{2})}$. For a characteristic temperature in our model, $T_{c}\sim 15$
keV, $\left\langle\gamma_{\pm}\right\rangle\ll 1$. thermal distributions in
the comoving frame, $\nu_{\rm 0,IC}$ satisfies the relationship (Rybicki &
Lightman, 1986):

$\kappa_{\rm es}=\bigg{(}\frac{m_{\rm e}c^{2}}{4kT}\bigg{)}\kappa_{\rm
ff}(\nu_{\rm 0,IC}).$ (17)

Substituting Eq. (17) into Eq. (• ‣ 3), one can derive

$\displaystyle x_{\rm 0,IC}\equiv\frac{h\nu_{\rm 0,IC}}{kT}\sim$
$\displaystyle 2.4\times 10^{17}(n_{\pm}m_{e})^{1/2}T^{-9/4}$
$\displaystyle\times[\overline{g}_{ff}(x_{\rm 0,IC})]^{1/2}.$ (18)

From Eqs. (• ‣ 3), (13) and (17), we have

$x_{\rm 0,IC}=\bigg{(}\frac{m_{\rm e}c^{2}}{4kT}\bigg{)}^{1/2}x_{\rm 0,CC}.$
(19)

The final observed flux can be calculated by combining Eqs. (12) and
(16),i.e.,

$\displaystyle F(\nu_{\rm obs},\Gamma,T,n_{\pm},\alpha,l_{0})$
$\displaystyle=F_{\nu_{\rm obs}}^{MB}(\Gamma,T,n_{\pm},l_{0})$
$\displaystyle\quad+F_{\nu_{\rm obs}}^{\rm IC}(\Gamma,T,\alpha,l_{0}),$ (20)

which can be used to directly fit to the observed data. Note that we assumed
an on-axis observer for Eq. (18) so the Dopper factor,
$D=1/[\Gamma(1-\beta\mathrm{cos}\theta)]$, is replaced with the buck Lorentz
factor $\Gamma$ of the pair plasma.

Considering the physical conditions, the priors and allowed ranges of the five
free parameters in Eq. (20) are set up as follows:

  * •

$n_{\pm}$: The number density of the $e^{\pm}$ in the emission region. Log-
uniformed distributed in range [$10^{24.5},10^{26.5}$].

  * •

$T$: The thermodynamic equilibrium temperature in the co-moving reference.
$kT$ is uniformed distributed in range [5.0, 30.0].

  * •

$\Gamma$: The bulk Lorentz factor of the pair plasma. Uniformed distributed in
range [35.0, 70.0]

  * •

$l_{0}$: The initial radius of the expanding fireball. Log-uniformed
distributed in range [$10^{4.0},10^{6.0}$]

  * •

$\alpha$: The index related to the IC intensity. Uniformed distributed in
range [0.0, 5.0] so $e^{-\alpha}$ is a significant number in range of
[0.007,1].

Figure 4: Time-integrated spectral fit of our model to the observed spectra of GRB 200415A between $T_{0}-0.005s$ and $T_{0}+0.20s$. Left: observed photon count spectra over-plotted with best-fit model; Middle: Deconvolved model-predicated photon spectrum; Right: corner diagram of the parameter constraints. Histograms show the 1-D likelihood distribution of the fitting parameters, and contours show the 2-D likelihood map constrained by the MCMC method. Red crosses mark the best-fit values, and contours represent the 1-,2-,3-$\sigma$ regions.  Table 1: The spectral fitting results of GRB 200415A Time Interval (s) |  | Flux |  |  | Model Parameters |  |  |
---|---|---|---|---|---|---|---|---
$t_{1}$ | $t_{2}$ | (erg cm-2s-1) |  $log_{10}(n_{\pm}/$cm${}^{-3})$ |  $kT$ (keV) | $\Gamma$ | $\alpha$ | $log_{10}(l_{0}/cm)$ | PGSTAT/dof
-0.005 | 0.20 | $2.30_{-0.48}^{+0.83}\times 10^{-3}$ | $25.40_{-0.005}^{+0.29}$ | $9.17_{-0.48}^{+5.41}$ | $56.09_{-17.86}^{+3.90}$ | $2.91_{-0.40}^{+0.60}$ | $4.66_{-0.09}^{+0.19}$ | 290.17/349
-0.005 | -0.001 | $0.01_{-0.01}^{+0.01}$ | $25.40_{-0.13}^{+0.24}$ | $6.11_{-0.12}^{+5.54}$ | $52.21_{-14.02}^{+8.08}$ | $2.51_{-0.17}^{+1.62}$ | $5.61_{-0.14}^{+0.13}$ | 253.4/349
-0.001 | 0.01 | $0.02_{-0.01}^{+0.01}$ | $26.04_{-0.09}^{+0.15}$ | $15.07_{-0.02}^{+0.42}$ | $55.09_{-0.10}^{+1.80}$ | $1.99_{-0.10}^{+0.04}$ | $4.87_{-0.05}^{+0.04}$ | 284.8/349
0.01 | 0.04 | $3.97_{-0.92}^{+1.70}\times 10^{-3}$ | $25.99_{-0.41}^{+0.42}$ | $12.60_{-0.51}^{+8.24}$ | $67.80_{-27.45}^{+2.16}$ | $4.99_{-0.98}^{+0.01}$ | $4.65_{-0.05}^{+0.29}$ | 245.9/349
0.04 | 0.12 | $8.58_{-1.98}^{+5.57}\times 10^{-4}$ | $25.70_{-0.02}^{+0.37}$ | $5.06_{-0.06}^{+7.90}$ | $49.53_{-10.65}^{+11.68}$ | $1.12_{-0.12}^{+3.48}$ | $5.12_{-0.09}^{+0.04}$ | 237.5/349
0.012 | 0.20 | $1.41_{-0.59}^{+12.74}\times 10^{-5}$ | $25.81_{-0.67}^{+0.45}$ | $5.10_{-0.10}^{+6.06}$ | $35.62_{-0.62}^{+17.23}$ | $4.43_{-1.26}^{+0.32}$ | 4.93${}_{-0.52}^{+0.16}$ | 178.1/349

We then collected the spectral data of GRB 200415A and performed the spectral
analysis on the main burst region between 0.005 and 0.20 s, following the
procedure of Yang et al. (2020). The data reduction follows the standard
procedure described in (Zhang et al., 2011, 2016, 2018). Both time-integrated
and time-dependent spectral analyses have been performed, between
$T_{0}-0.005s$ and $T_{0}+0.20s$. The spectral fitting slices are presented in
Table 1. By employing the self-developed spectral fitting package, McSpecfit
(Zhang et al., 2018), we were able to fit the spectral data using our model in
Eq. (20). The time-integrated fit fit is shown in Figure 4. The best-fit
parameters, along with the corresponding energy flux in each slice, are listed
in Table 1.

Figure 5: The equivalent lines of $\kappa_{\rm ff}$=$\kappa_{\rm es}$ for
different photon energies in the $n_{\pm}$–$kT$ parameter space. The region
above each line indicates where scattering becomes dominant. Constraints from
our spectral fitting of the observed data are marked with blue dots.

The values of the goodness of fit (PGSTAT/dof) in Table 1 indicate that our
model successfully explains the observation. The values of best-fit parameters
are overall consistent with the theoretical predictions as highlighted below:

  * •

$n_{\pm}$ is constrained to a relatively stable value between $10^{25}$ \-
$10^{26}$ cm-3, which is well consistent with the estimation in Eq. (3).

  * •

$kT$ clearly shows an intensity tracking pattern with a peak value at
$kT=15.7_{-0.02}^{+0.42}$ keV.

  * •

$\Gamma$ is constrained within the range of $\sim$ [50, 68], much smaller than
those in GRBs. Our results are also consistent with the lower limit (i.e.,
$\Gamma>6$) given by Roberts et al. (2021).

  * •

$l_{0}$ is stably constrained at $\sim 10^{5}$ cm, which is consistent with
model estimation in Eq. (2).

In Figure 5, we plot the best-fit values of the $n_{\pm}$ and $kT$ along the
critical lines defined by $\kappa_{\rm ff}(n_{\pm},kT,\nu)$=$\kappa_{\rm es}$,
above which scattering becomes dominated. Our results suggest a lower limit of
the photon energy for the scattering to take dominated effect (thus the
modified blackbody component can emerge), which roughly is $>1$ keV in the
comoving frame, or $>50$ keV in the observer frame by multiplying the $\Gamma$
values in Table 1.

##  4 Summary

In this Letter, we introduced a physical model which involves a fireball
bubble emerging from the magnetar surface due to violent neutron star
activities such as crust cracking or magnetic reconnection. We studied in
detail the evolution of such a fireball, including its expansion and
interaction with the magnetar $e^{\pm}$ wind. We found that high-energy
photons at the photosphere radius are Comptonized by the high-number-density
$e^{\pm}$ relativistic wind, which results in a modified blackbody spectrum
with a Wien tail. Through a direct Monte-Carlo fit, our physical model is
found well consistent with the observed spectra of GRB 200415A. Our results,
for the first time, confirm the physical origin of the magnetar giant flares
through first-principle theoretical calculations as well as the evidence of
data consistency.

We thank Di Xiao, Ken Chen, Qiao-Chu Li, Ze-Nan Liu, Jun Yang for helpful
discussions and constructive comments. Z.J.Z. appreciates the help from
Tianyue Li (School of Physics, NJU) and Shan Xiang (GXU) for their
illustration support. We acknowledge the use of public data from the Fermi
Science Support Center (FSSC). B.B.Z acknowledges support by Fundamental
Research Funds for the Central Universities (14380046), the National Key
Research and Development Programs of China (2018YFA0404204), the National
Natural Science Foundation of China (Grant Nos. 11833003, U2038105), the
science research grants from the China Manned Space Project with NO.CMS-
CSST-2021-B11, and the Program for Innovative Talents, Entrepreneur in
Jiangsu. Y.Z.M. is supported by the National Postdoctoral Program for
Innovative Talents (grant No. BX20200164).

## References

  * Atteia et al. (1987) Atteia, J. L., Boer, M., Hurley, K., et al. 1987, ApJ, 320, L105, doi: 10.1086/184984
  * Beloborodov (2010) Beloborodov, A. M. 2010, MNRAS, 407, 1033, doi: 10.1111/j.1365-2966.2010.16770.x
  * Bissaldi et al. (2020) Bissaldi, E., Briggs, M., Burns, E., et al. 2020, GRB Coordinates Network, 27587, 1
  * Boggs et al. (2007) Boggs, S. E., Zoglauer, A., Bellm, E., et al. 2007, The Astrophysical Journal, 661, 458, doi: 10.1086/516732
  * Duncan & Thompson (1992) Duncan, R. C., & Thompson, C. 1992, ApJ, 392, L9, doi: 10.1086/186413
  * Feroci et al. (2002) Feroci, M., Hurley, K., Duncan, R. C., & Thompson, C. 2002, Mem. Soc. Astron. Italiana, 73, 554
  * Golenetskii et al. (1984) Golenetskii, S. V., Ilinskii, V. N., & Mazets, E. P. 1984, Nature, 307, 41, doi: 10.1038/307041a0
  * Gourgouliatos et al. (2016) Gourgouliatos, K. N., Wood, T. S., & Hollerbach, R. 2016, Proceedings of the National Academy of Science, 113, 3944, doi: 10.1073/pnas.1522363113
  * Hurley et al. (1999) Hurley, K., Cline, T., Mazets, E., et al. 1999, Nature, 397, 41, doi: 10.1038/16199
  * Israel et al. (2005) Israel, G. L., Belloni, T., Stella, L., et al. 2005, ApJ, 628, L53, doi: 10.1086/432615
  * Israel et al. (2016) Israel, G. L., Esposito, P., Rea, N., et al. 2016, MNRAS, 457, 3448, doi: 10.1093/mnras/stw008
  * Kaspi & Beloborodov (2017) Kaspi, V. M., & Beloborodov, A. M. 2017, ARA&A, 55, 261, doi: 10.1146/annurev-astro-081915-023329
  * Katz (2020) Katz, J. I. 2020, MNRAS, 499, 2319, doi: 10.1093/mnras/staa3042
  * Kumar & Bošnjak (2020) Kumar, P., & Bošnjak, Ž. 2020, MNRAS, 494, 2385, doi: 10.1093/mnras/staa774
  * Landau & Lifshitz (1969) Landau, L. D., & Lifshitz, E. M. 1969, Statistical physics. Pt.1
  * Laros et al. (1986) Laros, J. G., Fenimore, E. E., Fikani, M. M., Klebesadel, R. W., & Barat, C. 1986, Nature, 322, 152, doi: 10.1038/322152a0
  * Lin et al. (2020) Lin, L., Zhang, C. F., Wang, P., et al. 2020, Nature, 587, 63, doi: 10.1038/s41586-020-2839-y
  * Mazets et al. (1979a) Mazets, E. P., Golenetskij, S. V., & Guryan, Y. A. 1979a, Soviet Astronomy Letters, 5, 343
  * Mazets et al. (1979b) Mazets, E. P., Golentskii, S. V., Ilinskii, V. N., Aptekar, R. L., & Guryan, I. A. 1979b, Nature, 282, 587, doi: 10.1038/282587a0
  * Meng et al. (2018) Meng, Y.-Z., Geng, J.-J., Zhang, B.-B., et al. 2018, ApJ, 860, 72. doi:10.3847/1538-4357/aac2d9
  * Meng et al. (2019) Meng, Y.-Z., Liu, L.-D., Wei, J.-J., et al. 2019, ApJ, 882, 26. doi:10.3847/1538-4357/ab30c7
  * Meng et al. (2021) Meng, Y.-Z., Geng, J.-J., & Wu, X.-F. 2021, arXiv:2107.04532
  * Mereghetti (2008) Mereghetti, S. 2008, A&A Rev., 15, 225, doi: 10.1007/s00159-008-0011-z
  * Mereghetti et al. (2015) Mereghetti, S., Pons, J. A., & Melatos, A. 2015, Space Sci. Rev., 191, 315, doi: 10.1007/s11214-015-0146-y
  * Meszaros & Rees (1992) Meszaros, P., & Rees, M. J. 1992, MNRAS, 257, 29P, doi: 10.1093/mnras/257.1.29P
  * Metzger et al. (2019) Metzger, B. D., Margalit, B., & Sironi, L. 2019, MNRAS, 485, 4091. doi:10.1093/mnras/stz700
  * Moffatt (1985) Moffatt, H. K. 1985, Journal of Fluid Mechanics, 159, 359, doi: 10.1017/S0022112085003251
  * Parker (1983a) Parker, E. N. 1983a, ApJ, 264, 635, doi: 10.1086/160636
  * Parker (1983b) —. 1983b, ApJ, 264, 642, doi: 10.1086/160637
  * Roberts et al. (2021) Roberts, O. J., Veres, P., Baring, M. G., et al. 2021, Nature, 589, 207, doi: 10.1038/s41586-020-03077-8
  * Ruderman & Sutherland (1975) Ruderman, M. A., & Sutherland, P. G. 1975, ApJ, 196, 51, doi: 10.1086/153393
  * Rybicki & Lightman (1986) Rybicki, G. B., & Lightman, A. P. 1986, Radiative Processes in Astrophysics
  * Strohmayer & Watts (2005) Strohmayer, T. E., & Watts, A. L. 2005, ApJ, 632, L111, doi: 10.1086/497911
  * Svinkin et al. (2021) Svinkin, D., Frederiks, D., Hurley, K., et al. 2021, Nature, 589, 211, doi: 10.1038/s41586-020-03076-9
  * Thompson & Duncan (1995) Thompson, C., & Duncan, R. C. 1995, MNRAS, 275, 255, doi: 10.1093/mnras/275.2.255
  * Thompson & Duncan (2001) —. 2001, ApJ, 561, 980, doi: 10.1086/323256
  * Wang et al. (2021) Wang, X. I., Zheng, X., Xiao, S., et al. 2021, arXiv:2107.10452
  * Yamazaki et al. (2006) Yamazaki, R., Ioka, K., Takahara, F., & Shibazaki, N. 2006, in Journal of Physics Conference Series, Vol. 31, Journal of Physics Conference Series, 99–102, doi: 10.1088/1742-6596/31/1/016
  * Yang et al. (2020) Yang, J., Chand, V., Zhang, B.-B., et al. 2020, ApJ, 899, 106, doi: 10.3847/1538-4357/aba745
  * Yang & Zhang (2015) Yang, Y.-P., & Zhang, B. 2015, ApJ, 815, 45, doi: 10.1088/0004-637X/815/1/45
  * Zhang et al. (2016) Zhang, B.-B., Uhm, Z. L., Connaughton, V., Briggs, M. S., & Zhang, B. 2016, ApJ, 816, 72, doi: 10.3847/0004-637X/816/2/72
  * Zhang et al. (2011) Zhang, B.-B., Zhang, B., Liang, E.-W., et al. 2011, ApJ, 730, 141, doi: 10.1088/0004-637X/730/2/141
  * Zhang et al. (2018) Zhang, B. B., Zhang, B., Castro-Tirado, A. J., et al. 2018, Nature Astronomy, 2, 69, doi: 10.1038/s41550-017-0309-8
  * Zhang (2020) Zhang, B. 2020, Nature, 587, 45, doi: doi:10.1038/s41586-020-2828-


# Tetragonal modular quotients $X_{0}^{+d}(N)$

Petar Orlić Petar Orlić
University of Zagreb
Bijenička Cesta 30
10000 Zagreb
Croatia<EMAIL_ADDRESS>
###### Abstract.

Let $N$ be a positive integer. For every $d\mid N$ such that $(d,N/d)=1$ there
exists an Atkin-Lehner involution $w_{d}$ of a modular curve $X_{0}(N)$. In
this paper we determine all quotient curves $X_{0}(N)/w_{d}$ whose
$\mathbb{Q}$-gonality is equal to $4$ and all quotient curves $X_{0}(N)/w_{d}$
whose $\mathbb{C}$-gonality is equal to $4$.

###### Key words and phrases:

Modular curves, Gonality

###### 1991 Mathematics Subject Classification:

11G18, 11G30, 14H30, 14H51

The author was supported by the project “Implementation of cutting-edge
research and its application as part of the Scientific Center of Excellence
for Quantum and Complex Systems, and Representations of Lie Algebras“,
PK.1.1.02, European Union, European Regional Development Fund and by the
Croatian Science Foundation under the project no. IP-2022-10-5008.

##  1\. Introduction

Let $C$ be a smooth projective curve over a field $k$. The $k$-gonality of
$C$, denoted by $\textup{gon}_{k}C$, is the least degree of a non-constant
$k$-rational morphism $f:C\to\mathbb{P}^{1}$. There exists an upper bound for
$\textup{gon}_{k}C$, linear in terms of the genus, see 3.4.

When $C$ is a modular curve, there also exists a linear lower bound for the
$\mathbb{C}$-gonality. This was first proved by Zograf [22]. The constant was
afterwards improved by Abramovich [1] and later by Kim and Sarnak in Appendix
2 to [15].

The gonality of the modular curve $X_{0}(N)$ and its quotients has been
extensively studied over the years. Ogg [18] determined all hyperelliptic
curves $X_{0}(N)$, Bars [2] determined all bielliptic curves $X_{0}(N)$,
Hasegawa and Shimura determined all trigonal curves $X_{0}(N)$ over
$\mathbb{C}$ and $\mathbb{Q}$, Jeon and Park determined all tetragonal curves
$X_{0}(N)$ over $\mathbb{C}$, and Najman and Orlić [16] determined all curves
$X_{0}(N)$ with $\mathbb{Q}$-gonality equal to $4$, $5$, or $6$, and also
determined the $\mathbb{Q}$ and $\mathbb{C}$-gonality for many other curves
$X_{0}(N)$.

Regarding the gonality of the quotients of the curve $X_{0}(N)$, Furumoto and
Hasegawa [6] determined all hyperelliptic quotients, and Hasegawa and Shimura
[9, 10, 11] determined all trigonal quotients over $\mathbb{C}$. Bars,
Gonzalez, and Kamel [3] determined all bielliptic quotients of $X_{0}(N)$ for
squarefree levels $N$, Jeon [12] determined all bielliptic quotients
$X_{0}^{+}(N)$, and Bars, Kamel, and Schweizer [4] determined all bielliptic
quotients of $X_{0}(N)$ for non-squarefree levels $N$, completing the
classification of bielliptic quotients.

The next logical step is to determine all tetragonal quotients of $X_{0}(N)$.
All tetragonal quotients $X_{0}^{+}(N)$ over $\mathbb{C}$ and $\mathbb{Q}$
were determined in [19]. Here, we will do the same for the quotients
$X_{0}^{+d}(N):=X_{0}(N)/w_{d}$ for $d\neq N$ (the notation $X_{0}^{+d}(N)$
was taken from [9]). We also determine all curves $X_{0}^{+d}(N)$ of genus $4$
that are trigonal over $\mathbb{Q}$, thus completing the classification of all
$\mathbb{Q}$-trigonal curves $X_{0}^{+d}(N)$, mainly done by Hasegawa and
Shimura [9].

Our main results are the following theorems. For expository reasons, we also
include the previously solved case $d=N$.

######  Theorem 1.1.

The curve $X_{0}^{+d}(N):=X_{0}(N)/w_{d}$ is of genus $4$ and has
$\mathbb{Q}$-gonality equal to $3$ if and only if

$\displaystyle(N,d)\in\\{$
$\displaystyle(66,33),(70,70),(74,37),(82,82),(86,43),(90,90),(108,4),(108,108),$
$\displaystyle(112,7),(117,117),(161,161),(173,173),(199,199),(251,251),(311,311)\\}.$

######  Theorem 1.2.

The curve $X_{0}^{+d}(N):=X_{0}(N)/w_{d}$ has $\mathbb{Q}$-gonality equal to
$4$ if and only if the pair $(N,d)$ is in the following table. In all cases
when the genus of the curve $X_{0}^{+d}(N)$ is not $4$ (all genus $4$ cases
are listed in 3.5), the $\mathbb{C}$-gonality is also equal to $4$.

Additionally, for $N=243,271$, the curve $X_{0}^{+}(N)$ is tetragonal over
$\mathbb{C}$, but not over $\mathbb{Q}$.

$N$ | $d$ | $N$ | $d$ | $N$ | $d$ | $N$ | $d$
---|---|---|---|---|---|---|---
$60$ |  $3$, $5$ | $66$ |  $2$, $3$, $22$ | $68$ | $17$ | $70$ |  $2$, $5$, $7$, $70$
$74$ | $2$ | $76$ | $4$ | $77$ | $11$ | $78$ |  $2$, $3$, $6$, $13$, $78$
$80$ | $5$ | $82$ |  $2$, $82$ | $84$ | $3,4,7,12,21,28$ | $85$ |  $5$, $17$
$88$ |  $8$, $11$ | $90$ | $2,5,9,10,18,45,90$ | $91$ | $7$ | $93$ |  $3$, $31$
$96$ | $3$ | $98$ | $2$ | $99$ | $9$ | $100$ | $25$
$102$ |  $2$, $3$, $17$, $51$, $102$ | $104$ |  $8$, $13$ | $105$ | $3,5,7,15,21,35,105$ | $106$ |  $2$,$53$,$106$
$108$ |  $27$, $108$ | $110$ | $2,5,10,11,22,55,110$ | $111$ |  $3$, $37$ | $112$ |  $16$, $112$
$114$ |  $2$, $3$, $19$, $38$, $114$ | $115$ |  $5$, $23$ | $116$ |  $4$, $29$ | $117$ |  $9$, $117$
$118$ |  $2$, $59$, $118$ | $120$ |  $5$, $8$, $15$, $24$, $40$, $120$ | $122$ |  $2$, $61$ | $123$ |  $3$, $41$, $123$
$124$ |  $4$, $31$, $124$ | $126$ | $2,7,9,14,18,63,126$ | $129$ |  $3$, $43$ | $130$ |  $2$, $10$, $13$, $26$, $65$
$132$ |  $4$, $11$, $44$ | $133$ |  $19$, $133$ | $134$ |  $2$, $67$, $134$ | $135$ |  $5$, $27$
$136$ |  $8$, $17$, $136$ | $138$ |  $3$, $6$, $23$, $69$, $138$ | $140$ |  $4$, $35$, $140$ | $141$ |  $3$, $47$, $141$
$142$ |  $2$, $71$, $142$ | $143$ |  $11$, $13$ | $144$ |  $9$, $16$, $144$ | $145$ |  $29$, $145$
$146$ |  $2$, $73$ | $147$ | $49$ | $148$ |  $4$, $148$ | $152$ | $152$
$153$ |  $9$, $17$ | $155$ |  $5$, $31$ | $156$ |  $4$, $39$, $156$ | $157$ | $157$
$158$ |  $2$, $79$, $158$ | $159$ |  $3$, $53$ | $160$ |  $32$, $160$ | $161$ |  $7$, $23$, $161$
$163$ | $163$ | $165$ |  $11$, $15$, $165$ | $166$ |  $2$, $83$, $166$ | $168$ |  $21$, $24$, $56$
$171$ |  $9$, $19$, $171$ | $173$ | $173$ | $175$ | $175$ | $176$ |  $11$, $16$, $176$
$177$ |  $3$, $59$, $177$ | $183$ | $183$ | $184$ |  $8$, $23$, $184$ | $185$ | $185$
$188$ |  $4$, $47$, $188$ | $190$ |  $5$, $10$, $19$, $95$ | $192$ | $192$ | $193$ | $193$
$194$ | $194$ | $195$ |  $5$, $39$, $195$ | $196$ | $4$ | $197$ | $197$
$199$ | $199$ | $200$ | $200$ | $203$ | $203$ | $205$ |  $5$, $41$, $205$
$206$ |  $2$, $103$, $206$ | $207$ |  $9$, $23$, $207$ | $209$ |  $11$, $19$, $209$ | $211$ | $211$
$213$ |  $3$, $71$, $213$ | $215$ |  $5$, $43$ | $221$ |  $13$, $17$, $221$ | $223$ | $223$
$224$ | $224$ | $229$ | $229$ | $241$ | $241$ | $251$ | $251$
$257$ | $257$ | $263$ | $263$ | $269$ | $269$ | $279$ |  $9$, $31$, $279$
$281$ | $281$ | $284$ |  $4$, $71$, $284$ | $287$ |  $7$, $41$, $287$ | $299$ |  $13$, $23$, $299$
$311$ | $311$ | $359$ | $359$ |  |  |  |

Interestingly, as we can see, it turns out that all curves $X_{0}^{+d}(N)$ for
$d\neq N$ that are $\mathbb{C}$-tetragonal are also $\mathbb{Q}$-tetragonal.
However, this property does not hold for curves $X_{0}^{+}(N)$ since for
$N=243,271$ the curve $X_{0}^{+}(N)$ is $\mathbb{C}$-tetragonal, but not
$\mathbb{Q}$-tetragonal.

We use similar methods to the ones used in [19] to determine the tetragonal
curves $X_{0}^{+d}(N)$. In Section 3, we give lower bounds to the
$\mathbb{Q}$-gonality by computing the gonality over finite fields. In Section
4, we give lower bound to the $\mathbb{C}$-gonality via Castelnuovo-Severi
inequality. In Section 5, we construct degree $4$ rational morphisms to
$\mathbb{P}^{1}$, either via quotient maps to curves
$X_{0}(N)/\left<w_{d},w_{d^{\prime}}\right>$ or using Magma. In Section 6, we
use the graded Betti numbers to disprove the existence of degree $4$ morphisms
to $\mathbb{P}^{1}$.

Note that for each level $N$ that is not a prime power, there are multiple
quotients $X_{0}^{+d}(N)$ that need to be checked. For example, for $N=210$
which has four different prime factors, there are $14$ such quotients (we can
exclude the curve $X_{0}^{+}(210)$ because it has already been solved).
Therefore, it can be hard to track whether all quotients have been solved.

For the reader’s convenience, at the end of the paper we put Tables 2, 3, 4.
In these tables, for each level $N$, we give the links to all propositions
used to solve the quotients at that level.

A lot of the results in this paper rely on Magma computations. The codes that
verify all computations in this paper can be found on

https://github.com/orlic1/gonality_X0_quotients.

All computations were performed on the Euler server at the Department of
Mathematics, University of Zagreb with a Intel Xeon W-2133 CPU running at
3.60GHz and with 64 GB of RAM.

##  2\. Acknowledgements

Many thanks to Nikola Adžaga and Philippe Michaud-Jacobs for permission to use
their Magma codes as templates. Additionally, code and data associated to the
paper [21] by Jeremy Rouse, Andrew V. Sutherland, and David Zureick-Brown was
used in 3.3. Their code can be found on

https://github.com/AndrewVSutherland/ell-adic-galois-
images/tree/209c2f888669785151174f472ea2c9eafb6daaa9.

I am also grateful to Filip Najman for his helpful comments and suggestions.

##  3\. $\mathbb{F}_{p}$-gonality

For a curve $C$ defined over $\mathbb{Q}$ and $p$ a prime of good reduction of
$C$, it is known that

$\textup{gon}_{\mathbb{F}_{p}}(C)\leq\textup{gon}_{\mathbb{Q}}(C).$

This is an important tool for determining the $\mathbb{Q}$-gonality because it
is generally much easier to find the gonality over finite fields. When working
with the modular curve $X_{0}(N)$ and its quotients, a very helpful result is
Ogg’s inequality [18, Theorem 3.1], stated in simpler form in [8, Lemma 3.1].

######  Lemma 3.1.

Let $p$ be a prime not dividing $N$. Then the number of
$\mathbb{F}_{p^{2}}$-points on the curve $X_{0}(N)$ is at least

$L_{p}(N):=\frac{p-1}{12}\psi(N)+2^{\omega(N)}.$

Here $\psi(N)$ is the index of the congruence subgroup $\Gamma_{0}(N)$, equal
to $N\prod_{q\mid N}(1+\frac{1}{q})$, and $\omega(N)$ is the number of
different prime divisors of $N$.

######  Lemma 3.2.

[16, Lemma 3.5] Let $C$ be a curve over $\mathbb{Q}$, $p$ a prime of good
reduction for $C$, and $q$ a power of $p$. Suppose
$\\#C(\mathbb{F}_{q})>d(q+1)$. Then $\textup{gon}_{\mathbb{Q}}(C)>d$.

If the quotient curve $X_{0}^{+d}(N)$ is tetragonal, then we have a rational
composition map $X_{0}(N)\to X_{0}^{+d}(N)\to\mathbb{P}^{1}$ of degree $8$.
Therefore, by 3.2 we must have

(1) $\displaystyle L_{p}(N)$ $\displaystyle\leq 8(p^{2}+1)$

for all primes $p\nmid N$. Also, notice that we can exclude all levels $N$
which are prime powers since in that case the only quotient of $X_{0}(N)$ is
the curve $X_{0}^{+}(N)$ which has been solved in [19]. From now on, we will
suppose that $N$ is not a prime power.

Now, similarily as in [8, Lemma 3.2], we get that for $N>420$ there exists a
prime $p$ for which this inequality does not hold. In this way we can also
eliminate

$\displaystyle N\in\\{$ $\displaystyle
255,260,266,273,276,280,282,285,286,290,292,294,296,304,$ $\displaystyle
306,308,310-318,320,322,324,326-328,330-334,336-340,$ $\displaystyle
342-354,356-370,372-376,378-390,392-419\\}.$

The following result is a direct application of 3.2.

######  Proposition 3.3.

The curve $X_{0}^{+d}(N)$ is not tetragonal over $\mathbb{Q}$ for $N=420$ and
all $15$ possible values of $d$.

###### Proof.

Using Magma, we calculate that the curve $X_{0}(420)$ has $1128$ points over
$\mathbb{F}_{11^{2}}$. 3.2 now tells us that the $\mathbb{Q}$-gonality of the
curve $X_{0}(420)$ is at least $10$. Therefore, the $\mathbb{Q}$-gonality of
all quotient curves $X_{0}^{+d}(420)$ is at least $5$. ∎

In the following results we will use Poonen’s [20, Proposition A.1.].

######  Proposition 3.4.

Let $X$ be a curve of genus $g$ over a field $k$.

  1. (i)

If $L$ is a field extension of $k$, then
$\textup{gon}_{L}(X)\leq\textup{gon}_{k}(X)$.

  2. (ii)

If $k$ is algebraically closed and $L$ is a field extension of $k$, then
$\textup{gon}_{L}(X)=\textup{gon}_{k}(X)$.

  3. (iii)

If $g\geq 2$, then $\textup{gon}_{k}(X)\leq 2g-2$.

  4. (iv)

If $g\geq 2$ and $X(k)\neq\emptyset$, then $\textup{gon}_{k}(X)\leq g$.

  5. (v)

If $k$ is algebraically closed, then $\textup{gon}_{k}(X)\leq\frac{g+3}{2}$.

  6. (vi)

If $\pi:X\to Y$ is a dominant $k$-rational map, then
$\textup{gon}_{k}(X)\leq\deg\pi\cdot\textup{gon}_{k}(Y)$.

  7. (vii)

If $\pi:X\to Y$ is a dominant $k$-rational map, then
$\textup{gon}_{k}(X)\geq\textup{gon}_{k}(Y)$.

Since all modular curves $X_{0}(N)$ and their quotients have at least one
rational cusp, this result implies that their $\mathbb{Q}$-gonalities are
bounded from above by their genus.

We can also compute the $\mathbb{F}_{p}$-gonality of a curve defined over
$\mathbb{Q}$ by checking the dimensions of Riemann-Roch spaces of degree $d$
effective $\mathbb{F}_{p}$-rational divisors.

######  Proposition 3.5.

The $\mathbb{Q}$-gonality of the genus $4$ curve $X_{0}^{+d}(N)$ is equal to
$4$ for the following values of $N$ and $d$:

$(N,d))$ | $p$ | $(N,d)$ | $p$ | $(N,d)$ | $p$ | $(N,d)$ | $p$
---|---|---|---|---|---|---|---
$(60,3)$ | $7$ | $(60,5)$ | $7$ | $(66,2)$ | $13$ | $(68,17)$ | $3$
$(70,5)$ | $17$ | $(74,2)$ | $3$ | $(76,4)$ | $5$ | $(77,11)$ | $3$
$(80,5)$ | $7$ | $(82,2)$ | $7$ | $(85,5)$ | $23$ | $(85,17)$ | $7$
$(88,8)$ | $5$ | $(91,7)$ | $11$ | $(93,3)$ | $5$ | $(98,2)$ | $11$
$(100,25)$ | $3$ | $(108,27)$ | $5$ | $(110,55)$ | $7$ | $(133,19)$ | $5$
$(145,29)$ | $11$ | $(177,59)$ | $5$ | $(188,47)$ | $3$ |  |

###### Proof.

Using Magma, we compute that there are no functions of degree $\leq 3$ in
$\mathbb{F}_{p}(X_{0}^{+d}(N))$. To do this, we check that every degree $3$
effective $\mathbb{F}_{p}$-rational divisor has Riemann-Roch dimension equal
to $1$ (the costant function is always in the Riemann-Roch space).

We can reduce the number of divisors that need to be checked by noting the
following: If there exists a function $f$ over a field $k$ of a certain degree
and if $c\in k$, then the function $g(x):=\frac{1}{f(x)-c}$ has the same
degree and its polar divisor contains a $k$-rational point.

On the other hand, all these curves are of genus $4$ and have at least one
rational cusp. By 3.4, this implies that their $\mathbb{Q}$-gonality is at
most $4$. ∎

######  Proposition 3.6.

The $\mathbb{Q}$-gonality of the curve $X_{0}^{+d}(N)$ at least $5$ for the
following values of $N$ and $d$:

$(N,d)$ | $p$ | $(N,d)$ | $p$ | $(N,d)$ | $p$ | $(N,d)$ | $p$ | $(N,d)$ | $p$
---|---|---|---|---|---|---|---|---|---
$(132,33)$ | $5$ | $(140,5)$ | $3$ | $(140,28)$ | $3$ | $(150,25)$ | $7$ | $(154,2)$ | $3$
$(154,7)$ | $3$ | $(154,11)$ | $3$ | $(154,14)$ | $5$ | $(154,22)$ | $5$ | $(154,77)$ | $3$
$(164,41)$ | $3$ | $(165,3)$ | $2$ | $(165,5)$ | $2$ | $(165,33)$ | $7$ | $(165,55)$ | $2$
$(168,3)$ | $5$ | $(168,7)$ | $5$ | $(168,8)$ | $5$ | $(170,2)$ | $7$ | $(170,5)$ | $3$
$(170,10)$ | $3$ | $(170,17)$ | $3$ | $(170,34)$ | $3$ | $(170,85)$ | $3$ | $(172,4)$ | $3$
$(180,4)$ | $7$ | $(180,5)$ | $7$ | $(180,9)$ | $7$ | $(180,20)$ | $7$ | $(180,36)$ | $7$
$(180,45)$ | $7$ | $(186,2)$ | $5$ | $(186,31)$ | $5$ | $(186,62)$ | $5$ | $(192,3)$ | $5$
$(192,64)$ | $5$ | $(195,13)$ | $2$ | $(198,11)$ | $5$ | $(198,22)$ | $5$ | $(198,99)$ | $5$
$(200,8)$ | $3$ | $(200,25)$ | $3$ | $(201,3)$ | $2$ | $(201,67)$ | $2$ | $(204,68)$ | $5$
$(208,13)$ | $3$ | $(210,35)$ | $11$ | $(212,4)$ | $3$ | $(212,53)$ | $5$ | $(216,8)$ | $5$
$(216,27)$ | $5$ | $(218,2)$ | $5$ | $(218,109)$ | $5$ | $(219,3)$ | $5$ | $(219,73)$ | $2$
$(220,55)$ | $3$ | $(224,7)$ | $5$ | $(225,25)$ | $2$ | $(226,2)$ | $3$ | $(226,113)$ | $3$
$(232,8)$ | $3$ | $(232,29)$ | $3$ | $(234,13)$ | $5$ | $(234,26)$ | $5$ | $(234,117)$ | $5$
$(235,5)$ | $7$ | $(235,47)$ | $3$ | $(237,3)$ | $2$ | $(232,79)$ | $7$ | $(240,80)$ | $11$
$(242,2)$ | $3$ | $(242,121)$ | $3$ | $(244,4)$ | $3$ | $(244,61)$ | $3$ | $(247,13)$ | $5$
$(247,19)$ | $5$ | $(250,125)$ | $3$ | $(252,63)$ | $5$ | $(253,11)$ | $2$ | $(253,23)$ | $3$
$(254,127)$ | $3$ | $(258,86)$ | $5$ | $(259,7)$ | $5$ | $(261,29)$ | $2$ | $(265,5)$ | $3$
$(265,53)$ | $3$ | $(268,4)$ | $3$ | $(268,67)$ | $3$ | $(272,16)$ | $3$ | $(274,137)$ | $3$
$(275,11)$ | $2$ | $(278,139)$ | $3$ | $(288,9)$ | $5$ | $(288,32)$ | $5$ | $(291,3)$ | $5$
$(297,11)$ | $2$ | $(298,149)$ | $3$ | $(301,7)$ | $3$ | $(301,43)$ | $3$ | $(302,151)$ | $3$
$(323,19)$ | $3$ | $(325,25)$ | $7$ | $(355,71)$ | $3$ |  |  |  |

###### Proof.

Similarily as in the previous proposition, we use Magma to compute that there
are no functions of degree $\leq 4$ in $\mathbb{F}_{p}(X_{0}^{+d}(N))$. ∎

Some computations in 3.6 were running for more than an hour, especially in the
higher genus cases. This approach is not feasible for the curves in the next
proposition which are all of high genus. For example, the curve
$X_{0}^{+6}(246)$ is of genus $20$ and the curve $X_{0}^{+3}(300)$ is of genus
$22$. Instead, we can prove that the quotient curve
$X_{0}(N)/\left<w_{d},w_{d^{\prime}}\right>$, which is of smaller genus, is
not tetragonal.

######  Proposition 3.7.

The $\mathbb{Q}$-gonality of the curve $X_{0}^{+d}(N)$ is at least $5$ for the
following values of $N$ and $d$:

$N$ | $d$ | $p$ | $Y$ | $N$ | $d$ | $p$ | $Y$
---|---|---|---|---|---|---|---
$228$ |  $3$, $19$, $57$ | $5$ | $X_{0}(228)/\left<w_{3},w_{19}\right>$ | $228$ |  $12$, $57$, $76$ | $5$ | $X_{0}(228)/\left<w_{12},w_{57}\right>$
$228$ |  $4$, $57$ | $5$ | $X_{0}(228)/\left<w_{4},w_{57}\right>$ | $240$ |  $3$, $5$, $15$ | $11$ | $X_{0}(240)/\left<w_{3},w_{5}\right>$
$240$ |  $3$, $16$, $48$ | $11$ | $X_{0}(240)/\left<w_{3},w_{16}\right>$ | $246$ |  $6$, $82$, $123$ | $5$ | $X_{0}(246)/\left<w_{6},w_{82}\right>$
$264$ |  $3$, $8$, $24$ | $5$ | $X_{0}(264)/\left<w_{3},w_{8}\right>$ | $264$ |  $3$, $11$, $33$ | $5$ | $X_{0}(264)/\left<w_{3},w_{11}\right>$
$264$ |  $8$, $11$, $88$ | $5$ | $X_{0}(264)/\left<w_{8},w_{11}\right>$ | $270$ |  $5$, $27$, $135$ | $7$ | $X_{0}(270)/\left<w_{5},w_{27}\right>$
$300$ |  $3$, $100$ | $7$ | $X_{0}(300)/\left<w_{3},w_{100}\right>$ | $300$ |  $12$, $25$ | $7$ | $X_{0}(300)/\left<w_{12},w_{25}\right>$
$309$ |  $3$, $103$ | $5$ | $X_{0}(309)/\left<w_{3},w_{103}\right>$ |  |  |  |

###### Proof.

Using Magma, we compute that there are no $\mathbb{F}_{p}$-rational functions
of degree $\leq 4$ from $Y$ to $\mathbb{P}^{1}$. in
$\mathbb{F}_{p}(X_{0}(N)/\left<w_{d},w_{d^{\prime}}\right>)$. Since there is a
rational degree $2$ quotient map $X_{0}^{+d}(N)\to Y$, 3.4(vii) tells us that
$\mathbb{F}_{p}$-gonality of $X_{0}^{+d}(N)$ is $\geq 5$. ∎

In Section 6, we will see that all curves of genus at least $10$ that are not
$\mathbb{Q}$-tetragonal are also not $\mathbb{C}$-tetragonal. Furthermore, all
such curves have been solved in this section.

##  4\. Castelnuovo-Severi inequality

######  Proposition 4.1 (Castelnuovo-Severi inequality).

Let $k$ be a perfect field, and let $X,\ Y,\ Z$ be curves over $k$. Let non-
constant morphisms $\pi_{Y}:X\rightarrow Y$ and $\pi_{Z}:X\rightarrow Z$ over
$k$ be given, and let their degrees be $m$ and $n$, respectively. Assume that
there is no morphism $X\rightarrow X^{\prime}$ of degree $>1$ through which
both $\pi_{Y}$ and $\pi_{Z}$ factor. Then the following inequality holds:

(2) $g(X)\leq m\cdot g(Y)+n\cdot g(Z)+(m-1)(n-1).$

In this statement of Castelnuovo-Severi inequality the hypothetical morphism
$X\to X^{\prime}$ is defined over $\overline{k}$. However, in [14, Theorem 12]
it was recently proved that we can suppose that this morphism is defined over
$k$.

Since $\mathbb{C}$ and $\mathbb{Q}$ are both perfect fields, we can use
Castelnuovo-Severi inequality to get lower bounds on both $\mathbb{C}$ and
$\mathbb{Q}$-gonalities.

######  Proposition 4.2.

The $\mathbb{C}$-gonality of the curve $X_{0}^{+d}(N)$ is at least $5$ for the
following values of $N$ and $d$. Here $g$ denotes the genus of the curve
$X_{0}^{+d}(N)$ and $g^{\prime}$ denotes the genus of the quotient curve
$X_{0}(N)/\left<w_{d},w_{d^{\prime}}\right>$.

Table 0. * $(N,d)$ | $g$ | $d^{\prime}$ | $g^{\prime}$ | $(N,d)$ | $g$ | $d^{\prime}$ | $g^{\prime}$ | $(N,d)$ | $g$ | $d^{\prime}$ | $g^{\prime}$
---|---|---|---|---|---|---|---|---|---|---|---
$(132,3)$ | $10$ | $44$ | $3$ | $(132,12)$ | $10$ | $11$ | $3$ | $(138,2)$ | $11$ | $23$ | $3$
$(138,46)$ | $11$ | $2$ | $3$ | $(140,7)$ | $10$ | $20$ | $3$ | $(150,2)$ | $10$ | $75$ | $3$
$(150,3)$ | $10$ | $50$ | $3$ | $(156,3)$ | $11$ | $13$ | $3$ | $(156,12)$ | $11$ | $52$ | $3$
$(156,13)$ | $12$ | $3$ | $3$ | $(156,52)$ | $12$ | $12$ | $3$ | $(174,2)$ | $14$ | $87$ | $3$
$(174,3)$ | $14$ | $29$ | $3$ | $(174,6)$ | $13$ | $58$ | $4$ | $(174,29)$ | $13$ | $3$ | $3$
$(174,58)$ | $14$ | $6$ | $4$ | $(182,2)$ | $13$ | $91$ | $4$ | $(182,7)$ | $13$ | $26$ | $4$
$(182,13)$ | $12$ | $14$ | $4$ | $(182,14)$ | $11$ | $26$ | $3$ | $(182,26)$ | $10$ | $14$ | $3$
$(182,91)$ | $10$ | $14$ | $3$ | $(183,61)$ | $10$ | $3$ | $3$ | $(186,3)$ | $14$ | $62$ | $4$
$(186,6)$ | $14$ | $62$ | $5$ | $(186,62)$ | $14$ | $6$ | $5$ | $(190,2)$ | $14$ | $95$ | $3$
$(190,38)$ | $14$ | $10$ | $3$ | $(195,3)$ | $13$ | $65$ | $3$ | $(195,15)$ | $13$ | $39$ | $3$
$(198,2)$ | $14$ | $99$ | $5$ | $(198,9)$ | $15$ | $11$ | $5$ | $(204,3)$ | $16$ | $68$ | $5$
$(204,4)$ | $15$ | $51$ | $5$ | $(204,12)$ | $16$ | $51$ | $5$ | $(204,17)$ | $16$ | $4$ | $6$
$(210,2)$ | $21$ | $35$ | $8$ | $(210,3)$ | $21$ | $35$ | $7$ | $(210,5)$ | $19$ | $7$ | $7$
$(210,6)$ | $19$ | $35$ | $6$ | $(210,7)$ | $21$ | $5$ | $7$ | $(210,10)$ | $21$ | $14$ | $6$
$(210,14)$ | $16$ | $10$ | $6$ | $(210,15)$ | $21$ | $21$ | $7$ | $(210,21)$ | $19$ | $15$ | $7$
$(210,30)$ | $21$ | $35$ | $8$ | $(210,42)$ | $21$ | $35$ | $8$ | $(210,70)$ | $21$ | $2$ | $8$
$(210,105)$ | $19$ | $3$ | $7$ | $(220,4)$ | $16$ | $55$ | $4$ | $(220,5)$ | $16$ | $11$ | $3$
$(220,11)$ | $13$ | $5$ | $4$ | $(220,20)$ | $16$ | $44$ | $4$ | $(220,44)$ | $13$ | $20$ | $4$
$(222,2)$ | $18$ | $111$ | $4$ | $(222,3)$ | $17$ | $37$ | $5$ | $(222,6)$ | $18$ | $74$ | $3$
$(222,37)$ | $18$ | $3$ | $5$ | $(222,74)$ | $13$ | $6$ | $3$ | $(222,111)$ | $10$ | $6$ | $3$
$(230,2)$ | $17$ | $115$ | $5$ | $(230,5)$ | $16$ | $46$ | $5$ | $(230,10)$ | $16$ | $23$ | $6$
$(230,23)$ | $17$ | $10$ | $6$ | $(230,46)$ | $15$ | $5$ | $5$ | $(230,115)$ | $14$ | $2$ | $5$
$(231,3)$ | $15$ | $77$ | $3$ | $(231,7)$ | $15$ | $33$ | $4$ | $(231,11)$ | $15$ | $21$ | $4$
$(231,21)$ | $13$ | $33$ | $4$ | $(231,33)$ | $13$ | $21$ | $4$ | $(231,77)$ | $11$ | $3$ | $3$
$(234,18)$ | $18$ | $117$ | $7$ | $(234,9)$ | $17$ | $26$ | $6$ | $(234,18)$ | $18$ | $26$ | $7$
$(238,2)$ | $17$ | $119$ | $3$ | $(238,7)$ | $17$ | $17$ | $3$ | $(238,14)$ | $17$ | $34$ | $3$
$(238,17)$ | $15$ | $7$ | $3$ | $(238,34)$ | $15$ | $14$ | $3$ | $(245,5)$ | $10$ | $49$ | $3$
$(246,2)$ | $19$ | $123$ | $7$ | $(246,3)$ | $20$ | $41$ | $7$ | $(246,6)$ | $20$ | $41$ | $7$
$(246,82)$ | $20$ | $2$ | $8$ | $(248,8)$ | $15$ | $31$ | $3$ | $(249,3)$ | $14$ | $83$ | $3$
$(250,2)$ | $14$ | $125$ | $5$ | $(252,4)$ | $17$ | $63$ | $5$ | $(252,7)$ | $19$ | $9$ | $7$
$(252,9)$ | $19$ | $7$ | $7$ | $(252,28)$ | $19$ | $36$ | $7$ | $(252,36)$ | $19$ | $28$ | $7$
$(254,2)$ | $16$ | $127$ | $4$ | $(258,2)$ | $20$ | $43$ | $8$ | $(258,3)$ | $20$ | $86$ | $7$
$(258,6)$ | $21$ | $86$ | $7$ | $(258,43)$ | $21$ | $2$ | $8$ | $(258,129)$ | $18$ | $6$ | $7$
$(259,37)$ | $12$ | $7$ | $4$ | $(261,9)$ | $13$ | $29$ | $4$ | $(262,2)$ | $16$ | $131$ | $4$
$(267,3)$ | $15$ | $89$ | $4$ | $(270,2)$ | $22$ | $135$ | $7$ | $(270,5)$ | $21$ | $27$ | $8$
$(270,10)$ | $22$ | $54$ | $7$ | $(270,27)$ | $22$ | $5$ | $8$ | $(270,54)$ | $19$ | $10$ | $7$
$(272,17)$ | $16$ | $16$ | $6$ | $(274,2)$ | $16$ | $137$ | $6$ | $(275,25)$ | $13$ | $11$ | $4$
$(278,2)$ | $17$ | $139$ | $5$ | $(291,97)$ | $16$ | $3$ | $6$ | $(295,5)$ | $15$ | $59$ | $3$
$(297,27)$ | $16$ | $11$ | $6$ | $(298,2)$ | $18$ | $149$ | $7$ | $(300,4)$ | $19$ | $75$ | $7$
$(300,75)$ | $19$ | $4$ | $7$ | $(302,2)$ | $19$ | $151$ | $5$ | $(303,3)$ | $17$ | $101$ | $3$
$(303,101)$ | $10$ | $3$ | $3$ | $(305,5)$ | $14$ | $61$ | $4$ | $(305,61)$ | $12$ | $5$ | $4$
$(319,11)$ | $15$ | $29$ | $4$ | $(319,29)$ | $12$ | $11$ | $4$ | $(321,3)$ | $18$ | $107$ | $4$
$(321,107)$ | $12$ | $3$ | $4$ | $(323,17)$ | $15$ | $19$ | $5$ | $(329,7)$ | $16$ | $47$ | $3$
$(329,47)$ | $11$ | $7$ | $3$ | $(335,5)$ | $16$ | $67$ | $4$ | $(335,67)$ | $17$ | $5$ | $4$
$(341,11)$ | $14$ | $31$ | $4$ | $(341,31)$ | $16$ | $11$ | $4$ | $(355,5)$ | $18$ | $71$ | $4$
$(371,7)$ | $17$ | $53$ | $5$ | $(371,53)$ | $18$ | $7$ | $5$ | $(377,13)$ | $16$ | $29$ | $5$
$(377,29)$ | $14$ | $13$ | $5$ | $(391,17)$ | $16$ | $23$ | $5$ | $(391,23)$ | $18$ | $17$ | $5$

###### Proof.

The results of [6] and [9] tell us that these curves $X_{0}^{+d}(N)$ are not
hyperelliptic nor trigonal over $\mathbb{C}$. If there existed a degree $4$
map from $X_{0}^{+d}(N)$ to $\mathbb{P}^{1}$, then, by the Castelnuovo-Severi
inequality, this map would have to factor through the quotient map
$X_{0}^{+d}(N)\to X_{0}(N)/\left<w_{d},w_{d^{\prime}}\right>$ and the quotient
curve $X_{0}(N)/\left<w_{d},w_{d^{\prime}}\right>$ would need to be elliptic
or hyperelliptic. However, we can again use [6] to eliminate this possibility.
∎

##  5\. Rational morphisms to $\mathbb{P}^{1}$

In most cases, when there exists a degree $4$ morphism
$X_{0}^{+d}(N)\to\mathbb{P}^{1}$, we can realise it via the quotient map to
the curve $X_{0}(N)/\left<w_{d},w_{d^{\prime}}\right>$, as the following two
propositions show.

######  Proposition 5.1.

The quotient curve $X_{0}(N)/\left<w_{d},w_{d^{\prime}}\right>$ is an elliptic
curve for the following values of $N,d,d^{\prime}$:

$N$ | $(d,d^{\prime})$ | $N$ | $(d,d^{\prime})$ | $N$ | $(d,d^{\prime})$ | $N$ | $(d,d^{\prime})$
---|---|---|---|---|---|---|---
$70$ | $(2,35)$ | $86$ | $(2,43)$ | $96$ | $(3,32)$ | $99$ | $(9,11)$
$105$ | $(3,35)$ | $110$ | $(2,5)$ | $111$ | $(3,37)$ | $118$ | $(2,59)$
$123$ | $(3,41)$ | $124$ | $(4,31)$ | $141$ | $(3,47)$ | $142$ | $(2,71)$
$143$ | $(11,13)$ | $145$ | $(5,29)$ | $155$ | $(5,31)$ | $159$ | $(3,53)$
$188$ | $(4,47)$ |  |  |  |  |  |

######  Proposition 5.2.

The quotient curve $X_{0}(N)/\left<w_{d},w_{d^{\prime}}\right>$ is a
hyperelliptic curve for the following values of $N,d,d^{\prime}$. Here $g$
denotes the genus of the curve $X_{0}(N)/\left<w_{d},w_{d^{\prime}}\right>$.

$N$ | $(d,d^{\prime})$ | $g$ | $N$ | $(d,d^{\prime})$ | $g$ | $N$ | $(d,d^{\prime})$ | $g$
---|---|---|---|---|---|---|---|---
$66$ | $(3,22)$ | $2$ | $70$ | $(7,10)$ | $2$ | $78$ | $(2,3)$ | $3$
$84$ | $(3,4),(7,12),(4,21),(3,28)$ | $2$ | $88$ | $(8,11)$ | $2$ | $90$ | $(2,45),(5,18),(9,10)$ | $2$
$93$ | $(3,31)$ | $2$ | $102$ | $(2,51),(3,17)$ | $2$ | $104$ | $(8,13)$ | $2$
$105$ | $(3,5),(3,7),(7,15)$ | $3$ | $106$ | $(2,53)$ | $2$ | $110$ | $(2,5),(2,11),(5,22)$ | $3$
$112$ | $(7,16)$ | $2$ | $114$ | $(2,19)$ | $3$ | $114$ | $(3,38)$ | $2$
$115$ | $(5,23)$ | $2$ | $116$ | $(4,29)$ | $2$ | $117$ | $(9,13)$ | $2$
$120$ | $(8,15),(15,24)$ | $2$ | $120$ | $(5,24)$ | $3$ | $122$ | $(2,61)$ | $2$
$126$ | $(2,63),(14,18)$ | $2$ | $126$ | $(7,9),(9,14)$ | $3$ | $129$ | $(3,43)$ | $2$
$130$ | $(10,26)$ | $2$ | $130$ | $(2,13)$ | $3$ | $132$ | $(4,11)$ | $2$
$133$ | $(7,19)$ | $2$ | $134$ | $(2,67)$ | $2$ | $135$ | $(5,27)$ | $2$
$136$ | $(8,17)$ | $3$ | $138$ | $(3,23),(6,23)$ | $2$ | $140$ | $(4,35)$ | $2$
$146$ | $(2,73)$ | $2$ | $147$ | $(3,49)$ | $2$ | $150$ | $(6,50)$ | $2$
$153$ | $(9,17)$ | $2$ | $156$ | $(4,39)$ | $2$ | $158$ | $(2,79)$ | $2$
$161$ | $(7,23)$ | $2$ | $165$ | $(11,15)$ | $3$ | $166$ | $(2,83)$ | $2$
$168$ | $(21,24)$ | $4$ | $171$ | $(9,19)$ | $3$ | $176$ | $(11,16)$ | $4$
$177$ | $(3,59)$ | $2$ | $184$ | $(8,23)$ | $2$ | $190$ | $(5,19)$ | $2$
$195$ | $(5,39)$ | $3$ | $205$ | $(5,41)$ | $2$ | $206$ | $(2,103)$ | $2$
$207$ | $(9,23)$ | $3$ | $209$ | $(11,19)$ | $2$ | $213$ | $(3,71)$ | $2$
$215$ | $(5,42)$ | $2$ | $221$ | $(13,17)$ | $2$ | $279$ | $(9,31)$ | $5$
$284$ | $(4,71)$ | $2$ | $287$ | $(7,41)$ | $2$ | $299$ | $(13,23)$ | $2$

###### Proof.

Every curve of genus $2$ is hyperelliptic and [6] gives us all hyperelliptic
quotients of genus $g\geq 3$. ∎

######  Proposition 5.3.

There exists a degree $3$ rational map from $X_{0}^{+d}(N)$ to
$\mathbb{P}^{1}$ for

$(N,d)\in\\{(66,33),(74,37),(86,43),(108,4),(112,7)\\}.$

###### Proof.

The curve $X_{0}^{+d}(N)$ is of genus $4$ and we can use the Magma function
Genus4GonalMap(C) to get the desired map. ∎

######  Proposition 5.4.

There exists a degree $4$ rational map from $X_{0}^{+d}(N)$ to
$\mathbb{P}^{1}$ for

$(N,d)\in\\{(144,9),(144,16),(148,4),(160,32),(196,4),(208,16),(217,31),\\}.$

###### Proof.

For $(N,d)\in\\{(144,9),(144,16),(196,4)\\}$ we found a function of degree $4$
by searching the Riemann-Roch spaces of divisors of the form
$P_{1}+P_{2}+P_{3}+P_{4}$, where $P_{i}\in X_{0}^{+d}(N)(\mathbb{Q})$.

In the other four cases we were not able to find a degree $4$ function whose
polar divisor is supported on rational points so we had to search for
quadratic points.

We searched for quadratic points by intersecting the curve $X_{0}^{+d}(N)$
with hyperplanes of the form

$b_{0}x_{0}+\ldots+b_{k}x_{k}=0,$

where $b_{0},\ldots,b_{k}\in\mathbb{Z}$ are coprime and chosen up to a certain
bound, a similar idea as in [5, Section 3.2]. We can improve this by noting
that, in a quadratic point $(x_{0},\ldots,x_{k})$, already its first three
coordinates must be linearly dependent over $\mathbb{Q}$. Therefore, it is
enough to check the hyperplanes

$b_{0}x_{0}+b_{1}x_{1}+b_{2}x_{2}=0.$

In all of these cases we found a function of degree $4$ lying in the Riemann-
Roch space of a divisor of the form $P_{1}+P_{2}+Q+\sigma(Q)$, where
$P_{1},P_{2}\in X_{0}^{+d}(N)(\mathbb{Q})$, and $Q$ is one of the quadratic
points we found. ∎

##  6\. Betti numbers

In this section we will prove that the remaining curves $X_{0}^{+d}(N)$ are
not $\mathbb{C}$-tetragonal. A helpful tool here is the Tower theorem [17,
Theorem 2.1]. We use its corollary [16, Corollary 4.6.] which says that for
curves of genus $\geq 10$, the existence of a map to $\mathbb{P}^{1}$ over
$\mathbb{C}$ is equivalent with the existence of a rational map to
$\mathbb{P}^{1}$.

Since in Section 3 we solved all cases when $g\geq 10$, the only curves we
need to look at are those of genus at most $9$. In order to bound the number
of levels we need to check, we can use the theorem by Kim and Sarnak,
mentioned in the Introduction.

######  Theorem 6.1 (Appendix 2 to [15]).

Let $X_{\Gamma}$ be the algebraic curve corresponding to a congruence subgroup
$\Gamma\subseteq\operatorname{SL}_{2}(\mathbb{Z})$ of index

$D_{\Gamma}=[\operatorname{SL}_{2}(\mathbb{Z}):\pm\Gamma].$

If $X_{\Gamma}$ is $d$-gonal, then $D_{\Gamma}\leq\frac{12000}{119}d$.

######  Corollary 6.2.

The curve $X_{0}^{+d}(N)$ is not $\mathbb{C}$-tetragonal for $N\geq 807$.

###### Proof.

Suppose $X_{0}^{+d}(N)$ is $\mathbb{C}$-tetragonal. Then $X_{0}(N)$ has a
degree $8$ map to $\mathbb{P}^{1}$. Since $-I\in\Gamma_{0}(N)$, we have that
$\psi(N)=D_{\Gamma_{0}(N)}\leq\frac{12000}{119}\cdot 8$ (here
$\psi(N)=N\prod_{q\mid N}(1+\frac{1}{q})$, as mentioned in 3.1). ∎

This leaves us with reasonably many cases that are not yet solved. The only
pairs $(N,d)$ we need to check are in the table below.

$(N,d)$ | $g(X_{0}^{+d}(N))$ | $(N,d)$ | $g(X_{0}^{+d}(N))$ | $(N,d)$ | $g(X_{0}^{+d}(N))$ | $(N,d)$ | $g(X_{0}^{+d}(N))$
---|---|---|---|---|---|---|---
$(102,6)$ | $8$ | $(102,34)$ | $8$ | $(114,6)$ | $9$ | $(114,57)$ | $8$
$(120,3)$ | $9$ | $(130,5)$ | $9$ | $(140,20)$ | $8$ | $(148,37)$ | $9$
$(152,8)$ | $8$ | $(152,19)$ | $9$ | $(154,7)$ | $9$ | $(154,77)$ | $9$
$(160,5)$ | $9$ | $(162,2)$ | $8$ | $(162,81)$ | $7$ | $(164,4)$ | $9$
$(172,43)$ | $9$ | $(174,87)$ | $8$ | $(175,7)$ | $8$ | $(175,25)$ | $8$
$(178,89)$ | $8$ | $(183,3)$ | $9$ | $(185,5)$ | $9$ | $(185,37)$ | $9$
$(187,11)$ | $9$ | $(187,17)$ | $7$ | $(196,49)$ | $9$ | $(202,101)$ | $9$
$(203,7)$ | $9$ | $(214,107)$ | $9$ | $(225,9)$ | $9$ | $(238,119)$ | $7$
$(245,49)$ | $9$ | $(248,31)$ | $9$ | $(249,83)$ | $8$ | $(262,131)$ | $9$
$(267,89)$ | $9$ | $(295,59)$ | $9$ |  |  |  |
Table 1. Curves $X_{0}^{+d}(N)$ of genus at most $9$ that are not
$\mathbb{Q}$-tetragonal.

We will use graded Betti numbers $\beta_{i,j}$ to disprove the existence of a
degree $4$ morphism to $\mathbb{P}^{1}$. We will follow the notation in [13,
Section 1.]. The results we mention can be found there and in [16, Section
3.1.].

######  Definition 6.3.

For a curve $X$ and divisor $D$ of degree $d$, $g_{d}^{r}$ is a subspace of
the Riemann-Roch space $L(D)$ such that $\dim V=r+1$.

Therefore, we want to determine whether $X_{0}^{+d}(N)$ has a $g_{4}^{1}$.
Green’s conjecture relates graded Betti numbers $\beta_{i,j}$ with the
existence of $g_{d}^{r}$.

######  Conjecture 6.4 (Green, [7]).

Let $X$ be a curve of genus $g$. Then $\beta_{p,2}\neq 0$ if and only if there
exists a divisor $D$ on $X$ of degree $d$ such that a subspace $g_{d}^{r}$ of
$L(D)$ satisfies $d\leq g-1$, $r=\ell(D)-1\geq 1$, and $d-2r\leq p$.

The ”if” part of this conjecture was proved in the same paper.

######  Theorem 6.5 (Green and Lazarsfeld, Appendix to [7]).

Let $X$ be a curve of genus $q$. If $\beta_{p,2}=0$, then there does not exist
a divisor $D$ on $X$ of degree $d$ such that a subspace $g_{d}^{r}$ of $L(D)$
satisfies $d\leq g-1$, $r\geq 1$, and $d-2r\leq p$.

######  Corollary 6.6.

Let $X$ be a curve of genus $g\geq 5$ with $\beta_{2,2}=0$. Then
$\textup{gon}_{\mathbb{C}}(X)\geq 5$.

######  Corollary 6.7.

The curve $X_{0}^{+d}(N)$ is not tetragonal for all $(N,d)$ in Table 1.

###### Proof.

For all these curves we compute $\beta_{2,2}=0$. ∎

##  7\. Proofs of the main theorems

###### Proof of 1.1.

Hasegawa and Shimura [9] already solved the cases when $g(X_{0}^{+}(N))\neq
4$. 3.5 and 5.3 solve the cases when the genus is equal to $4$. ∎

###### Proof of 1.2.

We can suppose that the genus of the curve $X_{0}^{+d}(N)$ is at least $4$,
otherwise the $\mathbb{Q}$-gonality is at most $3$ due to 3.4.

The results of [6] give us all hyperelliptic quotients of $X_{0}(N)$ and the
results of [9] give us all $\mathbb{C}$-trigonal curves $X_{0}^{+d}(N)$. There
are exactly $8$ cases when the curve $X_{0}^{+d}(N)$ is $\mathbb{C}$-trigonal
of genus $g\geq 5$, namely

$(N,d)\in\\{(117,13),(122,122),(146,146),(147,3),(162,162),(164,164),(181,181),(227,227)\\},$

and in these cases the Tower theorem implies that the $\mathbb{Q}$-gonality is
also equal to $3$.

For genus $4$ curves listed in the statement of the theorem, we used 3.5 to
prove that there are no degree $3$ rational maps to $\mathbb{P}^{1}$.
Therefore, the $\mathbb{Q}$-gonality of these curves must be equal to $4$.

We can now suppose that the curve $X_{0}^{+d}(N)$ is of genus $g\geq 5$ and is
not hyperelliptic nor trigonal over $\mathbb{C}$. For the curves listed in the
theorem, in Section 5 we find a rational degree $4$ map to $\mathbb{P}^{1}$.
In the remaining cases, we prove in Sections 3, 4, and 6 that there are no
degree $4$ maps to $\mathbb{P}^{1}$, and so
$\textup{gon}_{\mathbb{C}}X_{0}^{+d}(N)>4$ in these cases. ∎

##  8\. Summary of results

We summarize the results of this paper in the following table. For each value
of $N$, we give links to all results used to solve the curves $X_{0}^{+d}(N)$
for $d\neq N$. We skip the curves of genus at most $3$ and in the table we
write $g\leq 3$ when we want to say that all curves $X_{0}^{+d}(N)$ for that
level $N$ are of genus $g\leq 3$. We also skip the levels $N$ that are prime
powers.

$N$ | Results used | $N$ | Results used
---|---|---|---
$\leq 59$ | $g\leq 3$ | $60$ |  $g\leq 3$ for $d=4,15,20$, 3.5, [6] for $d=12$
$61$ |  | $62$ |  $g\leq 3$ for $d=31$, [6] for $d=2$
$63$ | $g\leq 3$ | $64$ |
$65$ | $g\leq 3$ | $66$ |  3.5, 5.2, 5.3, [6] for $d=6$
$67$ |  | $68$ |  $g\leq 3$ for $d=4$, 3.5
$69$ |  $g\leq 3$ for $d=23$, [6] for $d=3$ | $70$ |  $g\leq 3$ for $d=14,35$, 3.5, 5.1, 5.2, [6] for $d=10$
$71$ |  | $72$ | $g\leq 3$
$73$ |  | $74$ |  3.5, 5.3
$75$ | $g\leq 3$ | $76$ |  $g\leq 3$ for $d=19$, 3.5
$77$ |  $g\leq 3$ for $d=7$, 3.5 | $78$ |  5.2, [6] for $d=26$
$79$ |  | $80$ |  $g\leq 3$ for $d=16$, 3.5
$81$ |  | $82$ |  $g\leq 3$ for $d=41$, 3.5
$83$ |  | $84$ | 5.2
$85$ | 3.5 | $86$ |  5.1, 5.3
$87$ | [6] | $88$ |  3.5, 5.2
$89$ |  | $90$ | 5.2
$91$ |  $g\leq 3$ for $d=13$, 3.5 | $92$ | [6]
$93$ |  3.5, 5.2 | $94$ | [6]
$95$ | [6] | $96$ |  $g\leq 3$ for $d=32$, 5.1
$97$ |  | $98$ |  $g\leq 3$ for $d=49$, 3.5
$99$ |  $g\leq 3$ for $d=11$, 5.1 | $100$ |  $g\leq 3$ for $d=4$, 3.5
$101$ |  | $102$ |  5.2, 6.7
$103$ |  | $104$ | 5.2
$105$ |  5.1, 5.2 | $106$ | 5.2
$107$ |  | $108$ |  3.5, 5.3
$109$ |  | $110$ |  3.5, 5.1, 5.2
$111$ | 5.1 | $112$ |  5.2, 5.4
$113$ |  | $114$ |  5.2, 6.7
$115$ | 5.2 | $116$ | 5.2
$117$ |  5.2, [9] for $d=13$ | $118$ | 5.1
$119$ | [6] | $120$ |  5.2, 6.7
$121$ |  | $122$ | 5.2
$123$ | 5.1 | $124$ | 5.1
$125$ |  | $126$ | 5.2
$127$ |  | $128$ |
$129$ | 5.2 | $130$ |  5.2, 6.7
$131$ |  | $132$ |  3.6, 4.2, 5.2
$133$ |  3.5, 5.2 | $134$ | 5.2
$135$ | 5.2 | $136$ | 5.2
$137$ |  | $138$ |  4.2, 5.2
$139$ |  | $140$ |  3.6, 4.2, 5.2, 6.7

Table 2. Methods used for $N\leq 140$.

$N$ | Results used | $N$ | Results used | $N$ | Results used | $N$ | Results used
---|---|---|---|---|---|---|---
$141$ | 5.1 | $142$ | 5.1 | $143$ | 5.1 | $144$ | 5.4
$145$ |  3.5, 5.1 | $146$ | 5.2 | $147$ |  5.2, [9] for $d=3$ | $148$ |  5.4, 6.7
$149$ |  | $150$ |  3.6, 4.2, 5.2 | $151$ |  | $152$ | 6.7
$153$ | 5.2 | $154$ |  3.6, 6.7 | $155$ | 5.1 | $156$ |  4.2, 5.2
$157$ |  | $158$ | 5.2 | $159$ | 5.1 | $160$ |  3.6, 5.4
$161$ | 5.2 | $162$ | 6.7 | $163$ |  | $164$ |  3.6, 6.7
$165$ |  3.6, 5.2 | $166$ | 5.2 | $167$ | $168$ |  3.6, 5.2 |
$169$ |  | $170$ | 3.6 | $171$ | 5.2 | $172$ |  3.6, 6.7
$173$ |  | $174$ |  4.2, 6.7 | $175$ | 6.7 | $176$ | 5.2
$177$ |  3.5, 5.2 | $178$ |  4.2, 6.7 | $179$ |  | $180$ | 3.6
$181$ |  | $182$ | 4.2 | $183$ |  4.2, 6.7 | $184$ | 5.2
$185$ |  3.6, 6.7 | $186$ |  3.6, 4.2 | $187$ | 6.7 | $188$ |  3.5, 5.1
$189$ |  4.2, 6.7 | $190$ |  4.2, 5.2 | $191$ |  | $192$ | 3.6
$193$ |  | $194$ | 4.2 | $195$ |  3.6, 4.2, 5.2, 6.7 | $196$ |  5.4, 6.7
$197$ |  | $198$ |  3.6, 4.2 | $199$ |  | $200$ | 3.6
$201$ | 3.6 | $202$ |  4.2, 6.7 | $203$ |  4.2, 6.7 | $204$ |  3.6, 4.2
$205$ | 5.2 | $206$ | 5.2 | $207$ | 5.2 | $208$ |  3.6, 5.4
$209$ | 5.2 | $210$ |  3.6, 4.2 | $211$ |  | $212$ | 3.6
$213$ | 5.2 | $214$ |  4.2, 6.7 | $215$ | 5.2 | $216$ | 3.6
$217$ |  4.2, 5.4 | $218$ | 3.6 | $219$ | 3.6 | $220$ |  3.6, 4.2
$221$ | 5.2 | $222$ | 4.2 | $223$ |  | $224$ |  3.6, 4.2
$225$ |  3.6, 6.7 | $226$ | 3.6 | $227$ |  | $228$ | 3.7
$229$ |  | $230$ | 4.2 | $231$ | 4.2 | $232$ | 3.6
$233$ |  | $234$ |  3.6, 4.2 | $235$ | 3.6 | $236$ | 4.2
$237$ | 3.6 | $238$ |  4.2, 6.7 | $239$ |  | $240$ |  3.6, 3.7
$241$ |  | $242$ | 3.6 | $243$ |  | $244$ | 3.6
$245$ |  4.2, 6.7 | $246$ |  3.7, 4.2 | $247$ | 3.6 | $248$ |  4.2, 6.7
$249$ |  4.2, 6.7 | $250$ |  3.6, 4.2 | $251$ |  | $252$ |  3.6, 4.2
$253$ | 3.6 | $254$ |  3.6, 4.2 | $255$ | 3.2 | $256$ |
$257$ |  | $258$ |  3.6, 4.2 | $259$ |  3.6, 4.2 | $260$ | 3.2
$261$ |  3.6, 4.2 | $262$ |  4.2, 6.7 | $263$ |  | $264$ | 3.7
$265$ | 3.6 | $266$ | 3.2 | $267$ |  4.2, 6.7 | $268$ | 3.6
$269$ |  | $270$ |  3.7, 4.2 | $271$ |  | $272$ |  3.6, 4.2
$273$ |  3.6, 3.7 | $274$ |  3.6, 4.2 | $275$ |  3.6, 4.2 | $276$ | 3.2
$277$ |  | $278$ |  3.6, 4.2 | $279$ | 5.2 | $280$ | 3.2
$281$ |  | $282$ | 3.2 | $283$ |  | $284$ | 5.2
$285$ | 3.2 | $286$ | 3.2 | $287$ | 5.2 | $288$ | 3.6
$289$ |  | $290$ | 3.2 | $291$ |  3.6, 4.2 | $292$ | 3.2
$293$ |  | $294$ | 1 | $295$ |  4.2, 6.7 | $296$ | 3.2
$297$ |  3.6, 4.2 | $298$ |  3.6, 4.2 | $299$ | 5.2 | $300$ |  3.7, 4.2

Table 3. Methods used for $141\leq N\leq 300$.

$N$ | Results used | $N$ | Results used | $N$ | Results used | $N$ | Results used
---|---|---|---|---|---|---|---
$301$ | 3.6 | $302$ |  3.6, 4.2 | $303$ | 4.2 | $304$ | 3.2
$305$ | 4.2 | $309$ | 3.7 | $319$ | 4.2 | $321$ | 4.2
$323$ |  3.6, 4.2 | $325$ | 3.6 | $329$ | 4.2 | $335$ | 4.2
$341$ | 4.2 | $355$ |  3.6, 4.2 | $371$ | 4.2 | $377$ | 4.2
$391$ | 4.2 | $420$ | 3.3 |  |  |  |

Table 4. Methods used for $N\geq 300$. Levels $N$ eliminated by 3.2 and prime
powers are omitted.

## References

  * [1] D. Abramovich, A linear lower bound on the gonality of modular curves, Internat. Math. Res. Notices, (1996), pp. 1005–1011.
  * [2] F. Bars, Bielliptic modular curves, J. Number Theory, 76 (1999), pp. 154–165.
  * [3] F. Bars, J. González, and M. Kamel, Bielliptic quotient modular curves with $N$ square-free, J. Number Theory, 216 (2020), pp. 380–402.
  * [4] F. Bars, M. Kamel, and A. Schweizer, Bielliptic quotient modular curves of $X_{0}(N)$, Math. Comp., volume 92, number 340, March 2023, pages 895-929, (2022).
  * [5] J. Box, Quadratic points on modular curves with infinite Mordell-Weil group, Math. Comp., 90 (2021), pp. 321–343.
  * [6] M. Furumoto and Y. Hasegawa, Hyperelliptic Quotients of Modular Curves $X_{0}(N)$, Tokyo J. Math., 22 (1999), pp. 105 – 125.
  * [7] M. L. Green, Koszul cohomology and the geometry of projective varieties. Appendix: The nonvanishing of certain Koszul cohomology groups (by Mark Green and Robert Lazarsfeld), J. Differ. Geom., 19 (1984), pp. 125–167, 168–171.
  * [8] Y. Hasegawa and M. Shimura, Trigonal modular curves, Acta Arith., 88 (1999), pp. 129–140.
  * [9] , Trigonal modular curves $X_{0}^{+d}(N)$, Proc. Japan Acad., Ser. A, 75 (1999), pp. 172–175.
  * [10] , Trigonal modular curves $X_{0}^{*}(N)$, Proc. Japan Acad., Ser. A, 76 (2000), pp. 83–86.
  * [11] , Trigonal quotients of modular curves $X_{0}(N)$., Proc. Japan Acad., Ser. A, 82 (2006), pp. 15–17.
  * [12] D. Jeon, Bielliptic modular curves $X_{0}^{+}(N)$, J. Number Theory, 185 (2018), pp. 319–338.
  * [13] D. Jeon and E. Park, Tetragonal modular curves, Acta Arith., 120 (2005), pp. 307–312.
  * [14] M. Khawaja and S. Siksek, Primitive algebraic points on curves.  https://arxiv.org/abs/2306.17772, 2023.
  * [15] H. Kim, Functoriality for the exterior square of $\textup{GL}_{4}$ and the symmetric fourth of $\textup{GL}_{2}$. Appendix 2: Refined estimates towards the Ramanujan and Selberg conjectures (by Henry Kim and Peter Sarnak), J. Amer. Math. Soc., 16 (2002), p. 139–183.
  * [16] F. Najman and P. Orlić, Gonality of the modular curve $X_{0}(N)$, Math. Comp., (2023).
  * [17] K. V. Nguyen and M.-H. Saito, $d$-gonality of modular curves and bounding torsions.  https://arxiv.org/abs/alg-geom/9603024, 1996.
  * [18] A. P. Ogg, Hyperelliptic modular curves, Bull. Soc. Math. France, 102 (1974), pp. 449–462.
  * [19] P. Orlić, Tetragonal modular quotients $X_{0}^{+}(N)$.  https://arxiv.org/abs/2311.09955, 2023.
  * [20] B. Poonen, Gonality of modular curves in characteristic $p$, Math. Res. Lett., 14 (2007), pp. 691–701.
  * [21] J. Rouse, A. V. Sutherland, and D. Zureick-Brown, $\ell$ -adic images of galois for elliptic curves over $\mathbb{Q}$ (and an appendix with john voight), Forum Math. Sigma, 10 (2022), p. e62.
  * [22] P. G. Zograf, Small eigenvalues of automorphic laplacians in spaces of parabolic forms, J. Sov. Math., 36 (1987), pp. 106–114.