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 | $4^{8}\cdot8^{2}$ | 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: | 8N0 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}47&24\\126&23\end{bmatrix}$, $\begin{bmatrix}119&32\\152&159\end{bmatrix}$, $\begin{bmatrix}127&24\\106&139\end{bmatrix}$, $\begin{bmatrix}129&44\\142&141\end{bmatrix}$, $\begin{bmatrix}135&56\\2&37\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.0.i.2 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $64$ |
| Cyclic 168-torsion field degree: | $3072$ |
| 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, including 5 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4\cdot3^4\cdot7^2}\cdot\frac{x^{48}(5764801x^{16}-237180384x^{14}y^{2}+38423222208x^{12}y^{4}-746508317184x^{10}y^{6}+6734705886720x^{8}y^{8}-19744383246336x^{6}y^{10}+26878908284928x^{4}y^{12}-4388393189376x^{2}y^{14}+2821109907456y^{16})^{3}}{y^{4}x^{52}(7x^{2}-36y^{2})^{4}(7x^{2}+36y^{2})^{8}(49x^{4}-1512x^{2}y^{2}+1296y^{4})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.48.0-8.e.1.15 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 168.48.0-28.c.1.6 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.48.0-28.c.1.16 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.48.0-8.e.1.2 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.48.0-56.h.1.14 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.48.0-56.h.1.28 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.192.1-56.i.1.7 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-56.y.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-56.bc.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-56.bg.1.6 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-56.bu.1.6 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-56.by.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-56.cb.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-56.cd.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.fx.1.15 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.gd.2.7 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.hc.2.8 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.hi.1.16 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.me.2.16 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.mk.1.15 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.nk.1.13 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.nq.2.15 | $168$ | $2$ | $2$ | $1$ |
| 168.288.8-168.bz.2.64 | $168$ | $3$ | $3$ | $8$ |
| 168.384.7-168.cf.2.27 | $168$ | $4$ | $4$ | $7$ |