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Properties

Label 168.144.9.bkf.1
Level $168$
Index $144$
Genus $9$
Cusps $8$
$\Q$-cusps $0$

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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: $8$ (none of which are rational) Cusp widths $12^{4}\cdot24^{4}$ Cusp orbits $2^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: not computed
$\Q$-gonality: $4 \le \gamma \le 16$
$\overline{\Q}$-gonality: $4 \le \gamma \le 9$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24I9

Level structure

$\GL_2(\Z/168\Z)$-generators: $\begin{bmatrix}101&68\\32&91\end{bmatrix}$, $\begin{bmatrix}109&110\\80&167\end{bmatrix}$, $\begin{bmatrix}119&34\\92&53\end{bmatrix}$, $\begin{bmatrix}137&6\\132&43\end{bmatrix}$, $\begin{bmatrix}149&126\\84&163\end{bmatrix}$, $\begin{bmatrix}157&108\\156&41\end{bmatrix}$, $\begin{bmatrix}163&92\\56&149\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 168.288.9-168.bkf.1.1, 168.288.9-168.bkf.1.2, 168.288.9-168.bkf.1.3, 168.288.9-168.bkf.1.4, 168.288.9-168.bkf.1.5, 168.288.9-168.bkf.1.6, 168.288.9-168.bkf.1.7, 168.288.9-168.bkf.1.8, 168.288.9-168.bkf.1.9, 168.288.9-168.bkf.1.10, 168.288.9-168.bkf.1.11, 168.288.9-168.bkf.1.12, 168.288.9-168.bkf.1.13, 168.288.9-168.bkf.1.14, 168.288.9-168.bkf.1.15, 168.288.9-168.bkf.1.16, 168.288.9-168.bkf.1.17, 168.288.9-168.bkf.1.18, 168.288.9-168.bkf.1.19, 168.288.9-168.bkf.1.20, 168.288.9-168.bkf.1.21, 168.288.9-168.bkf.1.22, 168.288.9-168.bkf.1.23, 168.288.9-168.bkf.1.24, 168.288.9-168.bkf.1.25, 168.288.9-168.bkf.1.26, 168.288.9-168.bkf.1.27, 168.288.9-168.bkf.1.28, 168.288.9-168.bkf.1.29, 168.288.9-168.bkf.1.30, 168.288.9-168.bkf.1.31, 168.288.9-168.bkf.1.32, 168.288.9-168.bkf.1.33, 168.288.9-168.bkf.1.34, 168.288.9-168.bkf.1.35, 168.288.9-168.bkf.1.36, 168.288.9-168.bkf.1.37, 168.288.9-168.bkf.1.38, 168.288.9-168.bkf.1.39, 168.288.9-168.bkf.1.40, 168.288.9-168.bkf.1.41, 168.288.9-168.bkf.1.42, 168.288.9-168.bkf.1.43, 168.288.9-168.bkf.1.44, 168.288.9-168.bkf.1.45, 168.288.9-168.bkf.1.46, 168.288.9-168.bkf.1.47, 168.288.9-168.bkf.1.48, 168.288.9-168.bkf.1.49, 168.288.9-168.bkf.1.50, 168.288.9-168.bkf.1.51, 168.288.9-168.bkf.1.52, 168.288.9-168.bkf.1.53, 168.288.9-168.bkf.1.54, 168.288.9-168.bkf.1.55, 168.288.9-168.bkf.1.56, 168.288.9-168.bkf.1.57, 168.288.9-168.bkf.1.58, 168.288.9-168.bkf.1.59, 168.288.9-168.bkf.1.60, 168.288.9-168.bkf.1.61, 168.288.9-168.bkf.1.62, 168.288.9-168.bkf.1.63, 168.288.9-168.bkf.1.64
Cyclic 168-isogeny field degree: $64$
Cyclic 168-torsion field degree: $3072$
Full 168-torsion field degree: $1032192$

Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
24.72.4.cd.1 $24$ $2$ $2$ $4$ $0$
168.72.4.bu.1 $168$ $2$ $2$ $4$ $?$
168.72.5.o.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.yp.1 $168$ $2$ $2$ $17$
168.288.17.bbd.1 $168$ $2$ $2$ $17$
168.288.17.bda.1 $168$ $2$ $2$ $17$
168.288.17.bfu.1 $168$ $2$ $2$ $17$
168.288.17.coj.1 $168$ $2$ $2$ $17$
168.288.17.cry.1 $168$ $2$ $2$ $17$
168.288.17.cte.1 $168$ $2$ $2$ $17$
168.288.17.cwn.1 $168$ $2$ $2$ $17$
168.288.17.hex.1 $168$ $2$ $2$ $17$
168.288.17.hey.1 $168$ $2$ $2$ $17$
168.288.17.hfm.1 $168$ $2$ $2$ $17$
168.288.17.hfp.1 $168$ $2$ $2$ $17$
168.288.17.hhi.1 $168$ $2$ $2$ $17$
168.288.17.hhl.1 $168$ $2$ $2$ $17$
168.288.17.hhz.1 $168$ $2$ $2$ $17$
168.288.17.hia.1 $168$ $2$ $2$ $17$
168.288.17.kwf.1 $168$ $2$ $2$ $17$
168.288.17.kwf.2 $168$ $2$ $2$ $17$
168.288.17.kwh.1 $168$ $2$ $2$ $17$
168.288.17.kwh.2 $168$ $2$ $2$ $17$
168.288.17.kwj.1 $168$ $2$ $2$ $17$
168.288.17.kwj.2 $168$ $2$ $2$ $17$
168.288.17.kwl.1 $168$ $2$ $2$ $17$
168.288.17.kwl.2 $168$ $2$ $2$ $17$
168.288.17.kwn.1 $168$ $2$ $2$ $17$
168.288.17.kwn.2 $168$ $2$ $2$ $17$
168.288.17.kwp.1 $168$ $2$ $2$ $17$
168.288.17.kwp.2 $168$ $2$ $2$ $17$
168.288.17.kwr.1 $168$ $2$ $2$ $17$
168.288.17.kwr.2 $168$ $2$ $2$ $17$
168.288.17.kwt.1 $168$ $2$ $2$ $17$
168.288.17.kwt.2 $168$ $2$ $2$ $17$
168.288.17.kwv.1 $168$ $2$ $2$ $17$
168.288.17.kwv.2 $168$ $2$ $2$ $17$
168.288.17.kwx.1 $168$ $2$ $2$ $17$
168.288.17.kwx.2 $168$ $2$ $2$ $17$
168.288.17.kwz.1 $168$ $2$ $2$ $17$
168.288.17.kwz.2 $168$ $2$ $2$ $17$
168.288.17.kxb.1 $168$ $2$ $2$ $17$
168.288.17.kxb.2 $168$ $2$ $2$ $17$
168.288.17.kxd.1 $168$ $2$ $2$ $17$
168.288.17.kxd.2 $168$ $2$ $2$ $17$
168.288.17.kxf.1 $168$ $2$ $2$ $17$
168.288.17.kxf.2 $168$ $2$ $2$ $17$
168.288.17.kxh.1 $168$ $2$ $2$ $17$
168.288.17.kxh.2 $168$ $2$ $2$ $17$
168.288.17.kxj.1 $168$ $2$ $2$ $17$
168.288.17.kxj.2 $168$ $2$ $2$ $17$