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 | $4^{8}\cdot8^{2}$ | Cusp orbits | $2^{5}$ | ||
| 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: | 8N0 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}7&10\\148&173\end{bmatrix}$, $\begin{bmatrix}111&22\\200&299\end{bmatrix}$, $\begin{bmatrix}113&172\\200&233\end{bmatrix}$, $\begin{bmatrix}199&302\\236&219\end{bmatrix}$, $\begin{bmatrix}311&242\\172&129\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.48.0.be.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 |
|---|---|---|---|---|---|
| 8.48.0-8.e.2.13 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 156.48.0-156.c.1.6 | $156$ | $2$ | $2$ | $0$ | $?$ |
| 312.48.0-156.c.1.25 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.48.0-8.e.2.1 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.48.0-312.t.2.10 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.48.0-312.t.2.62 | $312$ | $2$ | $2$ | $0$ | $?$ |
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.be.2.9 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.da.2.10 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.dy.2.12 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.eg.2.7 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.iz.2.14 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.jh.2.10 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.jo.2.6 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.jw.2.14 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.mu.2.10 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.nc.2.14 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.nl.2.11 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.nt.2.9 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.pg.2.12 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.po.2.13 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.pt.2.13 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.px.2.12 | $312$ | $2$ | $2$ | $1$ |
| 312.288.8-312.dn.1.37 | $312$ | $3$ | $3$ | $8$ |
| 312.384.7-312.dj.1.31 | $312$ | $4$ | $4$ | $7$ |