Invariants
| Level: | $312$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $7 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $12^{12}$ | Cusp orbits | $4^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 12$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 7$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12B7 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}25&114\\82&95\end{bmatrix}$, $\begin{bmatrix}213&206\\74&59\end{bmatrix}$, $\begin{bmatrix}229&120\\42&1\end{bmatrix}$, $\begin{bmatrix}275&218\\240&235\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.144.7.bs.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $224$ |
| Cyclic 312-torsion field degree: | $10752$ |
| Full 312-torsion field degree: | $6709248$ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=19$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.144.4-24.g.1.11 | $24$ | $2$ | $2$ | $4$ | $1$ |
| 156.144.3-156.o.1.1 | $156$ | $2$ | $2$ | $3$ | $?$ |
| 312.144.3-156.o.1.3 | $312$ | $2$ | $2$ | $3$ | $?$ |
| 312.144.3-312.z.1.5 | $312$ | $2$ | $2$ | $3$ | $?$ |
| 312.144.3-312.z.1.6 | $312$ | $2$ | $2$ | $3$ | $?$ |
| 312.144.3-312.br.1.4 | $312$ | $2$ | $2$ | $3$ | $?$ |
| 312.144.3-312.br.1.14 | $312$ | $2$ | $2$ | $3$ | $?$ |
| 312.144.4-24.g.1.9 | $312$ | $2$ | $2$ | $4$ | $?$ |
| 312.144.4-312.br.1.5 | $312$ | $2$ | $2$ | $4$ | $?$ |
| 312.144.4-312.br.1.6 | $312$ | $2$ | $2$ | $4$ | $?$ |
| 312.144.4-312.cn.1.29 | $312$ | $2$ | $2$ | $4$ | $?$ |
| 312.144.4-312.cn.1.30 | $312$ | $2$ | $2$ | $4$ | $?$ |
| 312.144.4-312.co.1.27 | $312$ | $2$ | $2$ | $4$ | $?$ |
| 312.144.4-312.co.1.28 | $312$ | $2$ | $2$ | $4$ | $?$ |