Invariants
| Level: | $312$ | $\SL_2$-level: | $78$ | Newform level: | $1$ | ||
| Index: | $448$ | $\PSL_2$-index: | $224$ | ||||
| Genus: | $13 = 1 + \frac{ 224 }{12} - \frac{ 0 }{4} - \frac{ 8 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $2^{2}\cdot6^{2}\cdot26^{2}\cdot78^{2}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $8$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 24$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 13$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 78G13 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}127&213\\258&199\end{bmatrix}$, $\begin{bmatrix}189&133\\82&201\end{bmatrix}$, $\begin{bmatrix}306&163\\149&229\end{bmatrix}$, $\begin{bmatrix}308&19\\131&66\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.224.13.bm.2 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $12$ |
| Cyclic 312-torsion field degree: | $1152$ |
| Full 312-torsion field degree: | $4313088$ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=23$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 156.224.7-78.e.2.4 | $156$ | $2$ | $2$ | $7$ | $?$ |
| 312.224.5-312.a.2.10 | $312$ | $2$ | $2$ | $5$ | $?$ |
| 312.224.5-312.a.2.11 | $312$ | $2$ | $2$ | $5$ | $?$ |
| 312.224.7-312.c.1.7 | $312$ | $2$ | $2$ | $7$ | $?$ |
| 312.224.7-312.c.1.20 | $312$ | $2$ | $2$ | $7$ | $?$ |
| 312.224.7-78.e.2.2 | $312$ | $2$ | $2$ | $7$ | $?$ |