Invariants
| Level: | $312$ | $\SL_2$-level: | $8$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $2^{3}\cdot4$ | ||
| 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: | 8O0 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}3&212\\256&137\end{bmatrix}$, $\begin{bmatrix}53&176\\10&167\end{bmatrix}$, $\begin{bmatrix}61&264\\310&163\end{bmatrix}$, $\begin{bmatrix}167&172\\100&181\end{bmatrix}$, $\begin{bmatrix}177&184\\298&125\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.48.0.cq.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $112$ |
| Cyclic 312-torsion field degree: | $5376$ |
| Full 312-torsion field degree: | $20127744$ |
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 |
|---|---|---|---|---|---|
| 24.48.0-24.l.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 312.48.0-24.l.1.16 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 104.48.0-104.i.2.9 | $104$ | $2$ | $2$ | $0$ | $?$ |
| 312.48.0-104.i.2.5 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.48.0-312.u.1.6 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.48.0-312.u.1.25 | $312$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 312.192.1-312.bv.1.3 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.cg.1.6 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.ci.1.4 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.cp.1.12 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.dg.1.5 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.dh.1.5 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.di.1.7 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.dj.1.7 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.mg.1.2 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.mh.1.4 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.mi.1.4 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.mj.1.12 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.mw.1.2 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.mx.1.2 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.my.1.6 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.mz.1.6 | $312$ | $2$ | $2$ | $1$ |
| 312.288.8-312.on.2.33 | $312$ | $3$ | $3$ | $8$ |
| 312.384.7-312.jc.1.19 | $312$ | $4$ | $4$ | $7$ |