Invariants
| Level: | $168$ | $\SL_2$-level: | $8$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| 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: | 8O0 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}5&72\\22&23\end{bmatrix}$, $\begin{bmatrix}53&4\\34&141\end{bmatrix}$, $\begin{bmatrix}121&164\\58&13\end{bmatrix}$, $\begin{bmatrix}133&164\\120&115\end{bmatrix}$, $\begin{bmatrix}143&104\\42&47\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.48.0.bs.2 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $64$ |
| Cyclic 168-torsion field degree: | $1536$ |
| Full 168-torsion field degree: | $1548288$ |
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 |
|---|---|---|---|---|---|
| 8.48.0-8.e.2.11 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 168.48.0-8.e.2.16 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.48.0-168.t.2.36 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.48.0-168.t.2.53 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.48.0-168.x.1.28 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.48.0-168.x.1.29 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.192.1-168.ba.1.5 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.bf.1.3 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.cy.2.3 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.db.2.5 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.dy.2.2 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.dz.2.12 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.eg.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.eh.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.gm.2.4 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.gn.2.10 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.gu.1.6 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.gv.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.hc.1.7 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.hd.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.hk.2.1 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.hl.2.6 | $168$ | $2$ | $2$ | $1$ |
| 168.288.8-168.lx.1.16 | $168$ | $3$ | $3$ | $8$ |
| 168.384.7-168.gn.1.44 | $168$ | $4$ | $4$ | $7$ |