Invariants
| Level: | $168$ | $\SL_2$-level: | $12$ | ||||
| Index: | $32$ | $\PSL_2$-index: | $16$ | ||||
| Genus: | $0 = 1 + \frac{ 16 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $4\cdot12$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $4$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12B0 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}0&31\\73&117\end{bmatrix}$, $\begin{bmatrix}98&163\\99&19\end{bmatrix}$, $\begin{bmatrix}102&47\\163&131\end{bmatrix}$, $\begin{bmatrix}117&167\\137&150\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.16.0.a.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $96$ |
| Cyclic 168-torsion field degree: | $4608$ |
| Full 168-torsion field degree: | $4644864$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 10 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 16 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{18}}\cdot\frac{(x+y)^{16}(x^{4}+192y^{4})^{3}(x^{4}+1728y^{4})}{y^{12}x^{4}(x+y)^{16}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 84.16.0-6.a.1.4 | $84$ | $2$ | $2$ | $0$ | $?$ |
| 168.16.0-6.a.1.2 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.96.2-24.b.2.13 | $168$ | $3$ | $3$ | $2$ |
| 168.96.2-168.j.2.3 | $168$ | $3$ | $3$ | $2$ |
| 168.96.2-168.k.1.15 | $168$ | $3$ | $3$ | $2$ |
| 168.96.3-24.e.1.1 | $168$ | $3$ | $3$ | $3$ |
| 168.128.1-24.a.2.1 | $168$ | $4$ | $4$ | $1$ |
| 168.256.7-168.g.1.30 | $168$ | $8$ | $8$ | $7$ |