Invariants
| Level: | $312$ | $\SL_2$-level: | $8$ | 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 | $8^{24}$ | Cusp orbits | $2^{4}\cdot4^{2}\cdot8$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 5$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8A5 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}53&190\\124&139\end{bmatrix}$, $\begin{bmatrix}61&24\\12&25\end{bmatrix}$, $\begin{bmatrix}169&170\\120&203\end{bmatrix}$, $\begin{bmatrix}293&52\\244&261\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.192.5.hk.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $112$ |
| Cyclic 312-torsion field degree: | $2688$ |
| Full 312-torsion field degree: | $5031936$ |
Rational points
This modular curve has no $\Q_p$ points for $p=5$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.192.3-24.ba.1.1 | $24$ | $2$ | $2$ | $3$ | $0$ |
| 104.192.1-104.x.2.5 | $104$ | $2$ | $2$ | $1$ | $?$ |
| 312.192.1-104.x.2.11 | $312$ | $2$ | $2$ | $1$ | $?$ |
| 312.192.1-312.bm.1.2 | $312$ | $2$ | $2$ | $1$ | $?$ |
| 312.192.1-312.bm.1.17 | $312$ | $2$ | $2$ | $1$ | $?$ |
| 312.192.1-312.cy.1.1 | $312$ | $2$ | $2$ | $1$ | $?$ |
| 312.192.1-312.cy.1.24 | $312$ | $2$ | $2$ | $1$ | $?$ |
| 312.192.3-24.ba.1.6 | $312$ | $2$ | $2$ | $3$ | $?$ |
| 312.192.3-312.bk.2.13 | $312$ | $2$ | $2$ | $3$ | $?$ |
| 312.192.3-312.bk.2.25 | $312$ | $2$ | $2$ | $3$ | $?$ |
| 312.192.3-312.bl.1.2 | $312$ | $2$ | $2$ | $3$ | $?$ |
| 312.192.3-312.bl.1.3 | $312$ | $2$ | $2$ | $3$ | $?$ |
| 312.192.3-312.bx.1.1 | $312$ | $2$ | $2$ | $3$ | $?$ |
| 312.192.3-312.bx.1.2 | $312$ | $2$ | $2$ | $3$ | $?$ |