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Properties

Label 168.48.2.f.1
Level $168$
Index $48$
Genus $2$
Cusps $6$
$\Q$-cusps $6$

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Invariants

Level: $168$ $\SL_2$-level: $24$ Newform level: $1$
Index: $48$ $\PSL_2$-index:$48$
Genus: $2 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (all of which are rational) Cusp widths $2^{2}\cdot6^{2}\cdot8\cdot24$ Cusp orbits $1^{6}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: not computed
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $6$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24F2

Level structure

$\GL_2(\Z/168\Z)$-generators: $\begin{bmatrix}15&86\\32&33\end{bmatrix}$, $\begin{bmatrix}27&28\\44&19\end{bmatrix}$, $\begin{bmatrix}45&164\\136&5\end{bmatrix}$, $\begin{bmatrix}61&114\\6&121\end{bmatrix}$, $\begin{bmatrix}112&71\\141&14\end{bmatrix}$, $\begin{bmatrix}117&82\\146&109\end{bmatrix}$, $\begin{bmatrix}146&135\\95&166\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 168.96.2-168.f.1.1, 168.96.2-168.f.1.2, 168.96.2-168.f.1.3, 168.96.2-168.f.1.4, 168.96.2-168.f.1.5, 168.96.2-168.f.1.6, 168.96.2-168.f.1.7, 168.96.2-168.f.1.8, 168.96.2-168.f.1.9, 168.96.2-168.f.1.10, 168.96.2-168.f.1.11, 168.96.2-168.f.1.12, 168.96.2-168.f.1.13, 168.96.2-168.f.1.14, 168.96.2-168.f.1.15, 168.96.2-168.f.1.16, 168.96.2-168.f.1.17, 168.96.2-168.f.1.18, 168.96.2-168.f.1.19, 168.96.2-168.f.1.20, 168.96.2-168.f.1.21, 168.96.2-168.f.1.22, 168.96.2-168.f.1.23, 168.96.2-168.f.1.24, 168.96.2-168.f.1.25, 168.96.2-168.f.1.26, 168.96.2-168.f.1.27, 168.96.2-168.f.1.28, 168.96.2-168.f.1.29, 168.96.2-168.f.1.30, 168.96.2-168.f.1.31, 168.96.2-168.f.1.32, 168.96.2-168.f.1.33, 168.96.2-168.f.1.34, 168.96.2-168.f.1.35, 168.96.2-168.f.1.36, 168.96.2-168.f.1.37, 168.96.2-168.f.1.38, 168.96.2-168.f.1.39, 168.96.2-168.f.1.40, 168.96.2-168.f.1.41, 168.96.2-168.f.1.42, 168.96.2-168.f.1.43, 168.96.2-168.f.1.44, 168.96.2-168.f.1.45, 168.96.2-168.f.1.46, 168.96.2-168.f.1.47, 168.96.2-168.f.1.48, 168.96.2-168.f.1.49, 168.96.2-168.f.1.50, 168.96.2-168.f.1.51, 168.96.2-168.f.1.52, 168.96.2-168.f.1.53, 168.96.2-168.f.1.54, 168.96.2-168.f.1.55, 168.96.2-168.f.1.56, 168.96.2-168.f.1.57, 168.96.2-168.f.1.58, 168.96.2-168.f.1.59, 168.96.2-168.f.1.60, 168.96.2-168.f.1.61, 168.96.2-168.f.1.62, 168.96.2-168.f.1.63, 168.96.2-168.f.1.64
Cyclic 168-isogeny field degree: $16$
Cyclic 168-torsion field degree: $768$
Full 168-torsion field degree: $3096576$

Rational points

This modular curve has 6 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
$X_0(12)$ $12$ $2$ $2$ $0$ $0$

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

Covering curve Level Index Degree Genus
168.96.3.ds.2 $168$ $2$ $2$ $3$
168.96.3.ft.1 $168$ $2$ $2$ $3$
168.96.3.ih.2 $168$ $2$ $2$ $3$
168.96.3.ik.1 $168$ $2$ $2$ $3$
168.96.3.jo.1 $168$ $2$ $2$ $3$
168.96.3.jq.2 $168$ $2$ $2$ $3$
168.96.3.ka.1 $168$ $2$ $2$ $3$
168.96.3.kc.2 $168$ $2$ $2$ $3$
168.96.3.kr.1 $168$ $2$ $2$ $3$
168.96.3.ks.1 $168$ $2$ $2$ $3$
168.96.3.kv.1 $168$ $2$ $2$ $3$
168.96.3.kw.1 $168$ $2$ $2$ $3$
168.96.3.lh.2 $168$ $2$ $2$ $3$
168.96.3.li.1 $168$ $2$ $2$ $3$
168.96.3.ll.2 $168$ $2$ $2$ $3$
168.96.3.lm.1 $168$ $2$ $2$ $3$
168.96.3.pm.1 $168$ $2$ $2$ $3$
168.96.3.pm.2 $168$ $2$ $2$ $3$
168.96.3.pn.2 $168$ $2$ $2$ $3$
168.96.3.pn.4 $168$ $2$ $2$ $3$
168.96.3.pq.2 $168$ $2$ $2$ $3$
168.96.3.pq.4 $168$ $2$ $2$ $3$
168.96.3.pr.1 $168$ $2$ $2$ $3$
168.96.3.pr.2 $168$ $2$ $2$ $3$
168.96.3.pu.1 $168$ $2$ $2$ $3$
168.96.3.pu.2 $168$ $2$ $2$ $3$
168.96.3.pv.2 $168$ $2$ $2$ $3$
168.96.3.pv.4 $168$ $2$ $2$ $3$
168.96.3.py.1 $168$ $2$ $2$ $3$
168.96.3.py.3 $168$ $2$ $2$ $3$
168.96.3.pz.1 $168$ $2$ $2$ $3$
168.96.3.pz.2 $168$ $2$ $2$ $3$
168.96.3.qc.1 $168$ $2$ $2$ $3$
168.96.3.qc.2 $168$ $2$ $2$ $3$
168.96.3.qd.1 $168$ $2$ $2$ $3$
168.96.3.qd.3 $168$ $2$ $2$ $3$
168.96.3.qg.2 $168$ $2$ $2$ $3$
168.96.3.qg.4 $168$ $2$ $2$ $3$
168.96.3.qh.1 $168$ $2$ $2$ $3$
168.96.3.qh.2 $168$ $2$ $2$ $3$
168.96.3.qk.1 $168$ $2$ $2$ $3$
168.96.3.qk.2 $168$ $2$ $2$ $3$
168.96.3.ql.2 $168$ $2$ $2$ $3$
168.96.3.ql.4 $168$ $2$ $2$ $3$
168.96.3.qo.2 $168$ $2$ $2$ $3$
168.96.3.qo.4 $168$ $2$ $2$ $3$
168.96.3.qp.1 $168$ $2$ $2$ $3$
168.96.3.qp.2 $168$ $2$ $2$ $3$
168.144.7.czn.1 $168$ $3$ $3$ $7$