Invariants
| Level: | $168$ | $\SL_2$-level: | $8$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| 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: | 4E0 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}69&44\\56&111\end{bmatrix}$, $\begin{bmatrix}71&40\\101&39\end{bmatrix}$, $\begin{bmatrix}79&4\\101&151\end{bmatrix}$, $\begin{bmatrix}107&108\\74&143\end{bmatrix}$, $\begin{bmatrix}137&8\\84&11\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 28.12.0.g.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $64$ |
| Cyclic 168-torsion field degree: | $3072$ |
| Full 168-torsion field degree: | $6193152$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 631 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{7}\cdot\frac{(x+8y)^{12}(3x^{4}-784x^{2}y^{2}-6272xy^{3}-12544y^{4})^{3}}{x^{2}(x+8y)^{14}(x^{2}+14xy+56y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.12.0-4.c.1.5 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 168.12.0-4.c.1.5 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.192.5-28.k.1.9 | $168$ | $8$ | $8$ | $5$ |
| 168.504.16-28.s.1.6 | $168$ | $21$ | $21$ | $16$ |
| 168.48.0-56.be.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.be.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.bf.1.9 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.bf.1.11 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.bm.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.bm.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.bn.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.bn.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.72.2-84.s.1.17 | $168$ | $3$ | $3$ | $2$ |
| 168.96.1-84.k.1.23 | $168$ | $4$ | $4$ | $1$ |
| 168.48.0-168.cg.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.cg.1.13 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.ch.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.ch.1.14 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.co.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.co.1.14 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.cp.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.cp.1.13 | $168$ | $2$ | $2$ | $0$ |