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: | $\begin{bmatrix}37&24\\6&47\end{bmatrix}$, $\begin{bmatrix}47&32\\144&79\end{bmatrix}$, $\begin{bmatrix}55&80\\6&79\end{bmatrix}$, $\begin{bmatrix}91&12\\134&97\end{bmatrix}$, $\begin{bmatrix}149&104\\62&129\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.0.r.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, including 2 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4\cdot7^4}\cdot\frac{(2x-y)^{48}(1475789056x^{16}+843308032x^{14}y^{2}+90354432x^{12}y^{4}-4302592x^{10}y^{6}+8835680x^{8}y^{8}-21952x^{6}y^{10}+2352x^{4}y^{12}+112x^{2}y^{14}+y^{16})^{3}}{y^{8}x^{8}(2x-y)^{48}(14x^{2}-y^{2})^{4}(14x^{2}+y^{2})^{8}(196x^{4}+84x^{2}y^{2}+y^{4})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.48.0-8.e.2.14 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 168.48.0-8.e.2.9 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.48.0-56.i.2.17 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.48.0-56.i.2.32 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.48.0-56.l.1.6 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.48.0-56.l.1.17 | $168$ | $2$ | $2$ | $0$ | $?$ |
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.r.2.7 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-56.u.2.6 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-56.w.2.1 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-56.z.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-56.be.2.8 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-56.bf.2.4 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-56.bg.2.5 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-56.bh.2.8 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.im.2.5 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.in.2.2 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.iq.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.ir.1.5 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.js.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.jt.1.6 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.jw.2.6 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.jx.2.1 | $168$ | $2$ | $2$ | $1$ |
| 168.288.8-168.nh.1.36 | $168$ | $3$ | $3$ | $8$ |
| 168.384.7-168.hn.1.8 | $168$ | $4$ | $4$ | $7$ |