Properties

Label 168.48.0.eh.1
Level $168$
Index $48$
Genus $0$
Cusps $10$
$\Q$-cusps $0$

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Invariants

Level: $168$ $\SL_2$-level: $8$
Index: $48$ $\PSL_2$-index:$48$
Genus: $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$
Cusps: $10$ (none of which are rational) Cusp widths $2^{4}\cdot4^{2}\cdot8^{4}$ Cusp orbits $2^{3}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
$\Q$-gonality: $1 \le \gamma \le 2$
$\overline{\Q}$-gonality: $1$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8O0

Level structure

$\GL_2(\Z/168\Z)$-generators: $\begin{bmatrix}63&152\\61&95\end{bmatrix}$, $\begin{bmatrix}75&104\\50&159\end{bmatrix}$, $\begin{bmatrix}83&40\\143&63\end{bmatrix}$, $\begin{bmatrix}87&32\\79&157\end{bmatrix}$, $\begin{bmatrix}119&24\\139&133\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 168.96.0-168.eh.1.1, 168.96.0-168.eh.1.2, 168.96.0-168.eh.1.3, 168.96.0-168.eh.1.4, 168.96.0-168.eh.1.5, 168.96.0-168.eh.1.6, 168.96.0-168.eh.1.7, 168.96.0-168.eh.1.8, 168.96.0-168.eh.1.9, 168.96.0-168.eh.1.10, 168.96.0-168.eh.1.11, 168.96.0-168.eh.1.12, 168.96.0-168.eh.1.13, 168.96.0-168.eh.1.14, 168.96.0-168.eh.1.15, 168.96.0-168.eh.1.16
Cyclic 168-isogeny field degree: $32$
Cyclic 168-torsion field degree: $1536$
Full 168-torsion field degree: $3096576$

Models

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

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
8.24.0.ba.1 $8$ $2$ $2$ $0$ $0$
168.24.0.db.1 $168$ $2$ $2$ $0$ $?$
168.24.0.ed.2 $168$ $2$ $2$ $0$ $?$

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

Covering curve Level Index Degree Genus
168.144.8.sq.2 $168$ $3$ $3$ $8$
168.192.7.mj.2 $168$ $4$ $4$ $7$
168.384.23.nl.1 $168$ $8$ $8$ $23$