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Properties

Label 168.24.0.cz.1
Level $168$
Index $24$
Genus $0$
Cusps $6$
$\Q$-cusps $2$

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Invariants

Level: $168$ $\SL_2$-level: $8$
Index: $24$ $\PSL_2$-index:$24$
Genus: $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (of which $2$ are rational) Cusp widths $2^{4}\cdot8^{2}$ Cusp orbits $1^{2}\cdot2^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
$\Q$-gonality: $1$
$\overline{\Q}$-gonality: $1$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8G0

Level structure

$\GL_2(\Z/168\Z)$-generators: $\begin{bmatrix}59&64\\46&127\end{bmatrix}$, $\begin{bmatrix}109&48\\73&133\end{bmatrix}$, $\begin{bmatrix}157&72\\113&139\end{bmatrix}$, $\begin{bmatrix}159&64\\103&167\end{bmatrix}$, $\begin{bmatrix}159&160\\29&161\end{bmatrix}$, $\begin{bmatrix}167&32\\116&93\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 168.48.0-168.cz.1.1, 168.48.0-168.cz.1.2, 168.48.0-168.cz.1.3, 168.48.0-168.cz.1.4, 168.48.0-168.cz.1.5, 168.48.0-168.cz.1.6, 168.48.0-168.cz.1.7, 168.48.0-168.cz.1.8, 168.48.0-168.cz.1.9, 168.48.0-168.cz.1.10, 168.48.0-168.cz.1.11, 168.48.0-168.cz.1.12, 168.48.0-168.cz.1.13, 168.48.0-168.cz.1.14, 168.48.0-168.cz.1.15, 168.48.0-168.cz.1.16, 168.48.0-168.cz.1.17, 168.48.0-168.cz.1.18, 168.48.0-168.cz.1.19, 168.48.0-168.cz.1.20, 168.48.0-168.cz.1.21, 168.48.0-168.cz.1.22, 168.48.0-168.cz.1.23, 168.48.0-168.cz.1.24
Cyclic 168-isogeny field degree: $32$
Cyclic 168-torsion field degree: $1536$
Full 168-torsion field degree: $6193152$

Models

This modular curve is isomorphic to $\mathbb{P}^1$.

Rational points

This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
$X_0(8)$ $8$ $2$ $2$ $0$ $0$
168.12.0.s.1 $168$ $2$ $2$ $0$ $?$
168.12.0.bb.1 $168$ $2$ $2$ $0$ $?$

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

Covering curve Level Index Degree Genus
168.48.0.ed.1 $168$ $2$ $2$ $0$
168.48.0.ed.2 $168$ $2$ $2$ $0$
168.48.0.ee.1 $168$ $2$ $2$ $0$
168.48.0.ee.2 $168$ $2$ $2$ $0$
168.48.0.ef.1 $168$ $2$ $2$ $0$
168.48.0.ef.2 $168$ $2$ $2$ $0$
168.48.0.eg.1 $168$ $2$ $2$ $0$
168.48.0.eg.2 $168$ $2$ $2$ $0$
168.72.4.lf.1 $168$ $3$ $3$ $4$
168.96.3.mx.1 $168$ $4$ $4$ $3$
168.192.11.ld.1 $168$ $8$ $8$ $11$