Invariants
| Level: | $312$ | $\SL_2$-level: | $24$ | Newform level: | $1$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (none of which are rational) | Cusp widths | $2^{8}\cdot6^{8}\cdot8^{4}\cdot24^{4}$ | Cusp orbits | $2^{2}\cdot4^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 5$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24Z5 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}49&12\\7&307\end{bmatrix}$, $\begin{bmatrix}121&132\\31&97\end{bmatrix}$, $\begin{bmatrix}163&120\\52&89\end{bmatrix}$, $\begin{bmatrix}187&36\\224&257\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.192.5.rf.4 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $28$ |
| Cyclic 312-torsion field degree: | $2688$ |
| Full 312-torsion field degree: | $5031936$ |
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 |
|---|---|---|---|---|---|
| 24.192.1-24.cr.1.4 | $24$ | $2$ | $2$ | $1$ | $0$ |
| 312.192.1-24.cr.1.14 | $312$ | $2$ | $2$ | $1$ | $?$ |
| 312.192.1-312.sl.3.16 | $312$ | $2$ | $2$ | $1$ | $?$ |
| 312.192.1-312.sl.3.25 | $312$ | $2$ | $2$ | $1$ | $?$ |
| 312.192.1-312.sx.3.16 | $312$ | $2$ | $2$ | $1$ | $?$ |
| 312.192.1-312.sx.3.21 | $312$ | $2$ | $2$ | $1$ | $?$ |
| 312.192.3-312.hp.2.4 | $312$ | $2$ | $2$ | $3$ | $?$ |
| 312.192.3-312.hp.2.17 | $312$ | $2$ | $2$ | $3$ | $?$ |
| 312.192.3-312.hq.1.17 | $312$ | $2$ | $2$ | $3$ | $?$ |
| 312.192.3-312.hq.1.31 | $312$ | $2$ | $2$ | $3$ | $?$ |
| 312.192.3-312.si.3.16 | $312$ | $2$ | $2$ | $3$ | $?$ |
| 312.192.3-312.si.3.25 | $312$ | $2$ | $2$ | $3$ | $?$ |
| 312.192.3-312.su.3.16 | $312$ | $2$ | $2$ | $3$ | $?$ |
| 312.192.3-312.su.3.19 | $312$ | $2$ | $2$ | $3$ | $?$ |