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 | $2^{3}\cdot4$ | ||
| 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: | 8N0 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}13&216\\38&37\end{bmatrix}$, $\begin{bmatrix}137&108\\30&103\end{bmatrix}$, $\begin{bmatrix}147&136\\238&271\end{bmatrix}$, $\begin{bmatrix}155&64\\130&229\end{bmatrix}$, $\begin{bmatrix}213&20\\134&303\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.48.0.bn.2 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.i.2.2 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 104.48.0-104.i.2.25 | $104$ | $2$ | $2$ | $0$ | $?$ |
| 312.48.0-312.e.1.24 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.48.0-312.e.1.29 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.48.0-24.i.2.22 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.48.0-104.i.2.7 | $312$ | $2$ | $2$ | $0$ | $?$ |
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.co.1.10 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.cw.1.4 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.es.1.4 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.eu.1.10 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.il.1.4 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.ir.1.6 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.kg.1.4 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.km.1.4 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.mh.1.8 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.mn.1.9 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.oc.1.9 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.oi.1.8 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.pl.1.5 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.pn.1.8 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.qd.1.8 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.qe.1.3 | $312$ | $2$ | $2$ | $1$ |
| 312.288.8-312.eg.2.17 | $312$ | $3$ | $3$ | $8$ |
| 312.384.7-312.ds.1.29 | $312$ | $4$ | $4$ | $7$ |