Invariants
| Level: | $168$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot2\cdot4\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| 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: | 8I0 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}15&88\\136&111\end{bmatrix}$, $\begin{bmatrix}74&99\\27&10\end{bmatrix}$, $\begin{bmatrix}115&8\\100&79\end{bmatrix}$, $\begin{bmatrix}145&156\\30&151\end{bmatrix}$, $\begin{bmatrix}157&78\\26&97\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.24.0.bu.1 for the level structure with $-I$) |
| 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 infinitely many rational points, including 74 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^2}{7^2\cdot13^8}\cdot\frac{(7x+y)^{24}(62121073x^{8}-51323776x^{7}y-226174200x^{6}y^{2}-160864256x^{5}y^{3}+61244120x^{4}y^{4}+183318016x^{3}y^{5}+109082400x^{2}y^{6}+22378496xy^{7}+1811728y^{8})^{3}}{(x+2y)^{4}(7x+y)^{26}(7x^{2}-2y^{2})^{8}(42x^{2}-14xy-27y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.24.0-8.n.1.7 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 168.24.0-8.n.1.1 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.96.0-56.z.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-56.bc.2.1 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-56.bd.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-56.be.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-56.bg.2.5 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-56.bj.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-56.bl.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-56.bm.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.de.2.13 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dg.2.14 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.di.2.13 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dk.2.14 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.ea.2.15 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.ef.2.16 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.ej.2.15 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.em.1.16 | $168$ | $2$ | $2$ | $0$ |
| 168.144.4-168.no.1.35 | $168$ | $3$ | $3$ | $4$ |
| 168.192.3-168.ph.1.52 | $168$ | $4$ | $4$ | $3$ |
| 168.384.11-56.fc.1.20 | $168$ | $8$ | $8$ | $11$ |