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Properties

Label 168.96.3.qa.3
Level $168$
Index $96$
Genus $3$
Cusps $12$
$\Q$-cusps $2$

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Invariants

Level: $168$ $\SL_2$-level: $24$ Newform level: $1$
Index: $96$ $\PSL_2$-index:$96$
Genus: $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (of which $2$ are rational) Cusp widths $2^{4}\cdot6^{4}\cdot8^{2}\cdot24^{2}$ Cusp orbits $1^{2}\cdot2^{5}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: not computed
$\Q$-gonality: $2 \le \gamma \le 3$
$\overline{\Q}$-gonality: $2 \le \gamma \le 3$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24W3

Level structure

$\GL_2(\Z/168\Z)$-generators: $\begin{bmatrix}3&110\\166&119\end{bmatrix}$, $\begin{bmatrix}36&101\\25&56\end{bmatrix}$, $\begin{bmatrix}52&149\\79&42\end{bmatrix}$, $\begin{bmatrix}57&116\\56&141\end{bmatrix}$, $\begin{bmatrix}143&58\\98&135\end{bmatrix}$, $\begin{bmatrix}158&49\\135&148\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 168.192.3-168.qa.3.1, 168.192.3-168.qa.3.2, 168.192.3-168.qa.3.3, 168.192.3-168.qa.3.4, 168.192.3-168.qa.3.5, 168.192.3-168.qa.3.6, 168.192.3-168.qa.3.7, 168.192.3-168.qa.3.8, 168.192.3-168.qa.3.9, 168.192.3-168.qa.3.10, 168.192.3-168.qa.3.11, 168.192.3-168.qa.3.12, 168.192.3-168.qa.3.13, 168.192.3-168.qa.3.14, 168.192.3-168.qa.3.15, 168.192.3-168.qa.3.16, 168.192.3-168.qa.3.17, 168.192.3-168.qa.3.18, 168.192.3-168.qa.3.19, 168.192.3-168.qa.3.20, 168.192.3-168.qa.3.21, 168.192.3-168.qa.3.22, 168.192.3-168.qa.3.23, 168.192.3-168.qa.3.24, 168.192.3-168.qa.3.25, 168.192.3-168.qa.3.26, 168.192.3-168.qa.3.27, 168.192.3-168.qa.3.28, 168.192.3-168.qa.3.29, 168.192.3-168.qa.3.30, 168.192.3-168.qa.3.31, 168.192.3-168.qa.3.32
Cyclic 168-isogeny field degree: $16$
Cyclic 168-torsion field degree: $768$
Full 168-torsion field degree: $1548288$

Rational points

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

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
12.48.0.c.3 $12$ $2$ $2$ $0$ $0$
168.48.1.zu.1 $168$ $2$ $2$ $1$ $?$
168.48.2.g.2 $168$ $2$ $2$ $2$ $?$

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

Covering curve Level Index Degree Genus
168.192.5.lw.4 $168$ $2$ $2$ $5$
168.192.5.ro.4 $168$ $2$ $2$ $5$
168.192.5.sw.1 $168$ $2$ $2$ $5$
168.192.5.tf.4 $168$ $2$ $2$ $5$
168.192.5.wu.4 $168$ $2$ $2$ $5$
168.192.5.xb.4 $168$ $2$ $2$ $5$
168.192.5.xo.4 $168$ $2$ $2$ $5$
168.192.5.ya.4 $168$ $2$ $2$ $5$
168.192.5.zz.1 $168$ $2$ $2$ $5$
168.192.5.bag.4 $168$ $2$ $2$ $5$
168.192.5.bai.3 $168$ $2$ $2$ $5$
168.192.5.ban.4 $168$ $2$ $2$ $5$
168.192.5.bbv.4 $168$ $2$ $2$ $5$
168.192.5.bcc.4 $168$ $2$ $2$ $5$
168.192.5.bdc.4 $168$ $2$ $2$ $5$
168.192.5.bdh.4 $168$ $2$ $2$ $5$
168.288.13.buv.2 $168$ $3$ $3$ $13$