Invariants
| Level: | $168$ | $\SL_2$-level: | $42$ | Newform level: | $1$ | ||
| Index: | $448$ | $\PSL_2$-index: | $224$ | ||||
| Genus: | $15 = 1 + \frac{ 224 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $2$ are rational) | Cusp widths | $14^{4}\cdot42^{4}$ | Cusp orbits | $1^{2}\cdot3^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 15$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 15$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 42B15 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}26&49\\161&156\end{bmatrix}$, $\begin{bmatrix}81&70\\161&43\end{bmatrix}$, $\begin{bmatrix}84&79\\103&147\end{bmatrix}$, $\begin{bmatrix}98&55\\39&133\end{bmatrix}$, $\begin{bmatrix}167&133\\147&64\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.224.15.a.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $24$ |
| Cyclic 168-torsion field degree: | $1152$ |
| Full 168-torsion field degree: | $331776$ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(7)$ | $7$ | $16$ | $8$ | $0$ | $0$ |
| 24.16.0-24.a.1.2 | $24$ | $28$ | $28$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 84.224.6-21.a.1.16 | $84$ | $2$ | $2$ | $6$ | $?$ |
| 168.224.6-21.a.1.7 | $168$ | $2$ | $2$ | $6$ | $?$ |
| 24.16.0-24.a.1.2 | $24$ | $28$ | $28$ | $0$ | $0$ |