Invariants
| Level: | $312$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $6^{4}\cdot12^{4}$ | Cusp orbits | $2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12G3 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}63&14\\200&175\end{bmatrix}$, $\begin{bmatrix}79&196\\4&89\end{bmatrix}$, $\begin{bmatrix}99&202\\266&261\end{bmatrix}$, $\begin{bmatrix}147&62\\64&285\end{bmatrix}$, $\begin{bmatrix}295&290\\284&89\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.72.3.z.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $224$ |
| Cyclic 312-torsion field degree: | $21504$ |
| Full 312-torsion field degree: | $13418496$ |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.72.2-24.a.1.12 | $24$ | $2$ | $2$ | $2$ | $1$ |
| 312.72.2-24.a.1.10 | $312$ | $2$ | $2$ | $2$ | $?$ |
| 156.72.1-78.a.1.4 | $156$ | $2$ | $2$ | $1$ | $?$ |
| 312.72.1-78.a.1.2 | $312$ | $2$ | $2$ | $1$ | $?$ |
| 312.72.2-312.h.1.31 | $312$ | $2$ | $2$ | $2$ | $?$ |
| 312.72.2-312.h.1.32 | $312$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 312.288.7-312.e.1.3 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.e.1.9 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.o.1.3 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.o.1.24 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.bi.1.4 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.bi.1.16 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.bs.1.5 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.bs.1.6 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.pi.1.8 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.pi.1.16 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.pn.1.5 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.pn.1.8 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.qq.1.11 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.qq.1.14 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.qv.1.4 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.qv.1.16 | $312$ | $2$ | $2$ | $7$ |