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Properties

Label 168.144.9.sh.1
Level $168$
Index $144$
Genus $9$
Cusps $8$
$\Q$-cusps $2$

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Invariants

Level: $168$ $\SL_2$-level: $24$ Newform level: $1$
Index: $144$ $\PSL_2$-index:$144$
Genus: $9 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (of which $2$ are rational) Cusp widths $12^{4}\cdot24^{4}$ Cusp orbits $1^{2}\cdot2^{3}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: not computed
$\Q$-gonality: $3 \le \gamma \le 9$
$\overline{\Q}$-gonality: $3 \le \gamma \le 9$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24D9

Level structure

$\GL_2(\Z/168\Z)$-generators: $\begin{bmatrix}17&64\\124&33\end{bmatrix}$, $\begin{bmatrix}55&72\\60&127\end{bmatrix}$, $\begin{bmatrix}97&24\\84&127\end{bmatrix}$, $\begin{bmatrix}127&100\\72&35\end{bmatrix}$, $\begin{bmatrix}161&60\\144&107\end{bmatrix}$, $\begin{bmatrix}161&100\\148&159\end{bmatrix}$, $\begin{bmatrix}161&118\\52&19\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 168.288.9-168.sh.1.1, 168.288.9-168.sh.1.2, 168.288.9-168.sh.1.3, 168.288.9-168.sh.1.4, 168.288.9-168.sh.1.5, 168.288.9-168.sh.1.6, 168.288.9-168.sh.1.7, 168.288.9-168.sh.1.8, 168.288.9-168.sh.1.9, 168.288.9-168.sh.1.10, 168.288.9-168.sh.1.11, 168.288.9-168.sh.1.12, 168.288.9-168.sh.1.13, 168.288.9-168.sh.1.14, 168.288.9-168.sh.1.15, 168.288.9-168.sh.1.16, 168.288.9-168.sh.1.17, 168.288.9-168.sh.1.18, 168.288.9-168.sh.1.19, 168.288.9-168.sh.1.20, 168.288.9-168.sh.1.21, 168.288.9-168.sh.1.22, 168.288.9-168.sh.1.23, 168.288.9-168.sh.1.24, 168.288.9-168.sh.1.25, 168.288.9-168.sh.1.26, 168.288.9-168.sh.1.27, 168.288.9-168.sh.1.28, 168.288.9-168.sh.1.29, 168.288.9-168.sh.1.30, 168.288.9-168.sh.1.31, 168.288.9-168.sh.1.32, 168.288.9-168.sh.1.33, 168.288.9-168.sh.1.34, 168.288.9-168.sh.1.35, 168.288.9-168.sh.1.36, 168.288.9-168.sh.1.37, 168.288.9-168.sh.1.38, 168.288.9-168.sh.1.39, 168.288.9-168.sh.1.40, 168.288.9-168.sh.1.41, 168.288.9-168.sh.1.42, 168.288.9-168.sh.1.43, 168.288.9-168.sh.1.44, 168.288.9-168.sh.1.45, 168.288.9-168.sh.1.46, 168.288.9-168.sh.1.47, 168.288.9-168.sh.1.48, 168.288.9-168.sh.1.49, 168.288.9-168.sh.1.50, 168.288.9-168.sh.1.51, 168.288.9-168.sh.1.52, 168.288.9-168.sh.1.53, 168.288.9-168.sh.1.54, 168.288.9-168.sh.1.55, 168.288.9-168.sh.1.56, 168.288.9-168.sh.1.57, 168.288.9-168.sh.1.58, 168.288.9-168.sh.1.59, 168.288.9-168.sh.1.60, 168.288.9-168.sh.1.61, 168.288.9-168.sh.1.62, 168.288.9-168.sh.1.63, 168.288.9-168.sh.1.64
Cyclic 168-isogeny field degree: $64$
Cyclic 168-torsion field degree: $1536$
Full 168-torsion field degree: $1032192$

Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points.

Modular covers

The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank
$X_{\mathrm{ns}}^+(3)$ $3$ $48$ $48$ $0$ $0$
56.48.1.bi.1 $56$ $3$ $3$ $1$ $1$

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
24.72.4.z.1 $24$ $2$ $2$ $4$ $0$
56.48.1.bi.1 $56$ $3$ $3$ $1$ $1$
168.72.4.bi.2 $168$ $2$ $2$ $4$ $?$
168.72.5.d.1 $168$ $2$ $2$ $5$ $?$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus
168.288.17.bic.1 $168$ $2$ $2$ $17$
168.288.17.bwe.1 $168$ $2$ $2$ $17$
168.288.17.cir.1 $168$ $2$ $2$ $17$
168.288.17.cjr.1 $168$ $2$ $2$ $17$
168.288.17.dyq.1 $168$ $2$ $2$ $17$
168.288.17.dzm.1 $168$ $2$ $2$ $17$
168.288.17.ebd.1 $168$ $2$ $2$ $17$
168.288.17.ebz.1 $168$ $2$ $2$ $17$
168.288.17.fxl.1 $168$ $2$ $2$ $17$
168.288.17.fxt.1 $168$ $2$ $2$ $17$
168.288.17.gau.2 $168$ $2$ $2$ $17$
168.288.17.gbe.1 $168$ $2$ $2$ $17$
168.288.17.gca.1 $168$ $2$ $2$ $17$
168.288.17.gck.2 $168$ $2$ $2$ $17$
168.288.17.gdp.1 $168$ $2$ $2$ $17$
168.288.17.gdx.1 $168$ $2$ $2$ $17$
168.288.17.gmm.1 $168$ $2$ $2$ $17$
168.288.17.gmv.1 $168$ $2$ $2$ $17$
168.288.17.gpy.1 $168$ $2$ $2$ $17$
168.288.17.gqe.2 $168$ $2$ $2$ $17$
168.288.17.gre.2 $168$ $2$ $2$ $17$
168.288.17.grk.1 $168$ $2$ $2$ $17$
168.288.17.gsr.1 $168$ $2$ $2$ $17$
168.288.17.gsz.1 $168$ $2$ $2$ $17$
168.288.17.hiz.1 $168$ $2$ $2$ $17$
168.288.17.hjh.1 $168$ $2$ $2$ $17$
168.288.17.hmk.2 $168$ $2$ $2$ $17$
168.288.17.hmq.1 $168$ $2$ $2$ $17$
168.288.17.hnq.1 $168$ $2$ $2$ $17$
168.288.17.hnw.2 $168$ $2$ $2$ $17$
168.288.17.hpd.1 $168$ $2$ $2$ $17$
168.288.17.hpl.1 $168$ $2$ $2$ $17$
168.288.17.idl.1 $168$ $2$ $2$ $17$
168.288.17.idt.1 $168$ $2$ $2$ $17$
168.288.17.igu.1 $168$ $2$ $2$ $17$
168.288.17.ihe.2 $168$ $2$ $2$ $17$
168.288.17.iia.2 $168$ $2$ $2$ $17$
168.288.17.iik.1 $168$ $2$ $2$ $17$
168.288.17.ijt.1 $168$ $2$ $2$ $17$
168.288.17.ikb.1 $168$ $2$ $2$ $17$
168.288.17.iyg.1 $168$ $2$ $2$ $17$
168.288.17.izc.1 $168$ $2$ $2$ $17$
168.288.17.jas.1 $168$ $2$ $2$ $17$
168.288.17.jbo.1 $168$ $2$ $2$ $17$
168.288.17.jbw.1 $168$ $2$ $2$ $17$
168.288.17.jcw.1 $168$ $2$ $2$ $17$
168.288.17.jdq.1 $168$ $2$ $2$ $17$
168.288.17.jed.1 $168$ $2$ $2$ $17$