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: | $\begin{bmatrix}7&164\\32&273\end{bmatrix}$, $\begin{bmatrix}37&224\\280&187\end{bmatrix}$, $\begin{bmatrix}41&100\\82&153\end{bmatrix}$, $\begin{bmatrix}121&304\\252&199\end{bmatrix}$, $\begin{bmatrix}137&204\\174&205\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.48.0.cd.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $112$ |
| Cyclic 312-torsion field degree: | $10752$ |
| 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.i.1.28 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 104.48.0-104.i.2.20 | $104$ | $2$ | $2$ | $0$ | $?$ |
| 312.48.0-24.i.1.31 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.48.0-104.i.2.25 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.48.0-312.y.1.28 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.48.0-312.y.1.39 | $312$ | $2$ | $2$ | $0$ | $?$ |
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.cl.1.16 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.cn.1.14 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.ct.2.14 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.cv.2.16 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.es.2.11 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.et.2.13 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.eu.1.3 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.ev.1.8 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.fy.2.15 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.fz.2.16 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.ge.1.16 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.gf.1.15 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.hu.1.6 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.hv.1.4 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.ia.2.14 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.ib.2.9 | $312$ | $2$ | $2$ | $1$ |
| 312.288.8-312.my.1.64 | $312$ | $3$ | $3$ | $8$ |
| 312.384.7-312.hc.1.16 | $312$ | $4$ | $4$ | $7$ |