Invariants
| Level: | $312$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $4^{6}\cdot12^{6}$ | Cusp orbits | $2^{6}$ | ||
| 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: | 12K3 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}31&142\\102&53\end{bmatrix}$, $\begin{bmatrix}37&246\\102&67\end{bmatrix}$, $\begin{bmatrix}77&246\\46&121\end{bmatrix}$, $\begin{bmatrix}105&104\\170&309\end{bmatrix}$, $\begin{bmatrix}227&290\\196&39\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.96.3.eh.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $56$ |
| Cyclic 312-torsion field degree: | $5376$ |
| Full 312-torsion field degree: | $10063872$ |
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 |
|---|---|---|---|---|---|
| 24.96.1-24.by.1.20 | $24$ | $2$ | $2$ | $1$ | $0$ |
| 156.96.1-156.d.1.7 | $156$ | $2$ | $2$ | $1$ | $?$ |
| 312.96.1-156.d.1.5 | $312$ | $2$ | $2$ | $1$ | $?$ |
| 312.96.1-24.by.1.4 | $312$ | $2$ | $2$ | $1$ | $?$ |
| 312.96.1-312.dh.1.8 | $312$ | $2$ | $2$ | $1$ | $?$ |
| 312.96.1-312.dh.1.18 | $312$ | $2$ | $2$ | $1$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 312.384.5-312.kf.1.3 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.kf.2.6 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.kf.3.8 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.kf.4.6 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.ki.1.4 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.ki.2.8 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.ki.3.5 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.ki.4.4 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.pr.1.2 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.pr.2.7 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.pr.3.7 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.pr.4.8 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.pu.1.4 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.pu.2.8 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.pu.3.6 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.pu.4.2 | $312$ | $2$ | $2$ | $5$ |