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: | $10$ (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: | $\begin{bmatrix}81&140\\100&23\end{bmatrix}$, $\begin{bmatrix}85&120\\160&5\end{bmatrix}$, $\begin{bmatrix}93&100\\2&35\end{bmatrix}$, $\begin{bmatrix}107&80\\88&113\end{bmatrix}$, $\begin{bmatrix}153&104\\116&131\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.0.x.1 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{12}\cdot5^8\cdot7^2}\cdot\frac{(4x-3y)^{48}(692224x^{8}-3067904x^{7}y+13447168x^{6}y^{2}-38610432x^{5}y^{3}+57796480x^{4}y^{4}-44540608x^{3}y^{5}+31808448x^{2}y^{6}-25296936xy^{7}+11354329y^{8})^{3}(19369984x^{8}-37761024x^{7}y+41545728x^{6}y^{2}-50903552x^{5}y^{3}+57796480x^{4}y^{4}-33784128x^{3}y^{5}+10295488x^{2}y^{6}-2055256xy^{7}+405769y^{8})^{3}}{(4x-3y)^{48}(8x^{2}-7y^{2})^{8}(8x^{2}-32xy+7y^{2})^{4}(8x^{2}-7xy+7y^{2})^{8}(3648x^{4}-3584x^{3}y+2352x^{2}y^{2}-3136xy^{3}+2793y^{4})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.48.0-8.h.1.6 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 168.48.0-8.h.1.6 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.48.0-56.h.1.23 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.48.0-56.h.1.25 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.48.0-56.i.1.17 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.48.0-56.i.1.31 | $168$ | $2$ | $2$ | $0$ | $?$ |
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.e.2.2 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-56.k.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-56.p.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-56.v.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-56.bs.1.6 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-56.bt.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-56.by.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-56.bz.1.7 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.kq.1.5 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.kr.1.15 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.ks.1.15 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.kt.1.3 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.pc.2.11 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.pd.2.13 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.pm.2.13 | $168$ | $2$ | $2$ | $1$ |
| 168.192.1-168.pn.2.7 | $168$ | $2$ | $2$ | $1$ |
| 168.288.8-168.ov.2.24 | $168$ | $3$ | $3$ | $8$ |
| 168.384.7-168.jk.2.24 | $168$ | $4$ | $4$ | $7$ |