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Properties

Label 168.72.5.bx.1
Level $168$
Index $72$
Genus $5$
Cusps $4$
$\Q$-cusps $2$

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Invariants

Level: $168$ $\SL_2$-level: $24$ Newform level: $1$
Index: $72$ $\PSL_2$-index:$72$
Genus: $5 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (of which $2$ are rational) Cusp widths $12^{2}\cdot24^{2}$ Cusp orbits $1^{2}\cdot2$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: not computed
$\Q$-gonality: $2 \le \gamma \le 5$
$\overline{\Q}$-gonality: $2 \le \gamma \le 5$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24A5

Level structure

$\GL_2(\Z/168\Z)$-generators: $\begin{bmatrix}41&151\\124&139\end{bmatrix}$, $\begin{bmatrix}73&3\\0&151\end{bmatrix}$, $\begin{bmatrix}77&54\\124&49\end{bmatrix}$, $\begin{bmatrix}99&46\\128&27\end{bmatrix}$, $\begin{bmatrix}101&37\\148&139\end{bmatrix}$, $\begin{bmatrix}127&59\\56&125\end{bmatrix}$, $\begin{bmatrix}131&18\\132&35\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 168.144.5-168.bx.1.1, 168.144.5-168.bx.1.2, 168.144.5-168.bx.1.3, 168.144.5-168.bx.1.4, 168.144.5-168.bx.1.5, 168.144.5-168.bx.1.6, 168.144.5-168.bx.1.7, 168.144.5-168.bx.1.8, 168.144.5-168.bx.1.9, 168.144.5-168.bx.1.10, 168.144.5-168.bx.1.11, 168.144.5-168.bx.1.12, 168.144.5-168.bx.1.13, 168.144.5-168.bx.1.14, 168.144.5-168.bx.1.15, 168.144.5-168.bx.1.16, 168.144.5-168.bx.1.17, 168.144.5-168.bx.1.18, 168.144.5-168.bx.1.19, 168.144.5-168.bx.1.20, 168.144.5-168.bx.1.21, 168.144.5-168.bx.1.22, 168.144.5-168.bx.1.23, 168.144.5-168.bx.1.24, 168.144.5-168.bx.1.25, 168.144.5-168.bx.1.26, 168.144.5-168.bx.1.27, 168.144.5-168.bx.1.28, 168.144.5-168.bx.1.29, 168.144.5-168.bx.1.30, 168.144.5-168.bx.1.31, 168.144.5-168.bx.1.32
Cyclic 168-isogeny field degree: $64$
Cyclic 168-torsion field degree: $3072$
Full 168-torsion field degree: $2064384$

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
$X_{\mathrm{ns}}^+(3)$ $3$ $24$ $24$ $0$ $0$
56.24.1.n.1 $56$ $3$ $3$ $1$ $1$

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
12.36.2.p.1 $12$ $2$ $2$ $2$ $0$
56.24.1.n.1 $56$ $3$ $3$ $1$ $1$
168.36.0.gb.1 $168$ $2$ $2$ $0$ $?$
168.36.3.c.1 $168$ $2$ $2$ $3$ $?$

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

Covering curve Level Index Degree Genus
168.144.9.chw.1 $168$ $2$ $2$ $9$
168.144.9.cif.1 $168$ $2$ $2$ $9$
168.144.9.cri.1 $168$ $2$ $2$ $9$
168.144.9.crq.1 $168$ $2$ $2$ $9$
168.144.9.csm.1 $168$ $2$ $2$ $9$
168.144.9.ctf.1 $168$ $2$ $2$ $9$
168.144.9.cvc.1 $168$ $2$ $2$ $9$
168.144.9.cvv.1 $168$ $2$ $2$ $9$
168.144.9.cxl.1 $168$ $2$ $2$ $9$
168.144.9.cxp.1 $168$ $2$ $2$ $9$
168.144.9.cyc.1 $168$ $2$ $2$ $9$
168.144.9.cyk.1 $168$ $2$ $2$ $9$
168.144.9.czh.1 $168$ $2$ $2$ $9$
168.144.9.czn.1 $168$ $2$ $2$ $9$
168.144.9.dac.1 $168$ $2$ $2$ $9$
168.144.9.dak.1 $168$ $2$ $2$ $9$
168.144.9.dbn.1 $168$ $2$ $2$ $9$
168.144.9.dbv.1 $168$ $2$ $2$ $9$
168.144.9.dda.1 $168$ $2$ $2$ $9$
168.144.9.ddq.1 $168$ $2$ $2$ $9$
168.144.9.dfd.1 $168$ $2$ $2$ $9$
168.144.9.dfl.1 $168$ $2$ $2$ $9$
168.144.9.dgo.1 $168$ $2$ $2$ $9$
168.144.9.dhe.1 $168$ $2$ $2$ $9$
168.144.9.dja.1 $168$ $2$ $2$ $9$
168.144.9.djq.1 $168$ $2$ $2$ $9$
168.144.9.dlh.1 $168$ $2$ $2$ $9$
168.144.9.dlp.1 $168$ $2$ $2$ $9$
168.144.9.dmw.1 $168$ $2$ $2$ $9$
168.144.9.dnm.1 $168$ $2$ $2$ $9$
168.144.9.dpb.1 $168$ $2$ $2$ $9$
168.144.9.dpj.1 $168$ $2$ $2$ $9$
168.144.9.dqg.1 $168$ $2$ $2$ $9$
168.144.9.dqo.1 $168$ $2$ $2$ $9$
168.144.9.drl.1 $168$ $2$ $2$ $9$
168.144.9.drp.1 $168$ $2$ $2$ $9$
168.144.9.dsc.1 $168$ $2$ $2$ $9$
168.144.9.dsk.1 $168$ $2$ $2$ $9$
168.144.9.dth.1 $168$ $2$ $2$ $9$
168.144.9.dtl.1 $168$ $2$ $2$ $9$
168.144.9.dud.1 $168$ $2$ $2$ $9$
168.144.9.duq.1 $168$ $2$ $2$ $9$
168.144.9.dwl.1 $168$ $2$ $2$ $9$
168.144.9.dwy.1 $168$ $2$ $2$ $9$
168.144.9.dyw.1 $168$ $2$ $2$ $9$
168.144.9.dze.1 $168$ $2$ $2$ $9$
168.144.9.eab.1 $168$ $2$ $2$ $9$
168.144.9.eai.1 $168$ $2$ $2$ $9$