Invariants
| Level: | $312$ | $\SL_2$-level: | $4$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4G0 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}5&200\\192&115\end{bmatrix}$, $\begin{bmatrix}181&300\\250&175\end{bmatrix}$, $\begin{bmatrix}303&106\\70&57\end{bmatrix}$, $\begin{bmatrix}311&52\\288&113\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.24.0.d.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $224$ |
| Cyclic 312-torsion field degree: | $21504$ |
| Full 312-torsion field degree: | $40255488$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.24.0-8.a.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 104.24.0-8.a.1.4 | $104$ | $2$ | $2$ | $0$ | $?$ |
| 156.24.0-156.b.1.8 | $156$ | $2$ | $2$ | $0$ | $?$ |
| 312.24.0-156.b.1.3 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.24.0-312.b.1.5 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.24.0-312.b.1.14 | $312$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 312.144.4-312.k.1.16 | $312$ | $3$ | $3$ | $4$ |
| 312.192.3-312.dy.1.11 | $312$ | $4$ | $4$ | $3$ |