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: | $10$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8O0 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}95&176\\142&303\end{bmatrix}$, $\begin{bmatrix}139&32\\174&257\end{bmatrix}$, $\begin{bmatrix}165&244\\292&259\end{bmatrix}$, $\begin{bmatrix}235&224\\186&287\end{bmatrix}$, $\begin{bmatrix}237&116\\52&189\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.48.0.ce.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 infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
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 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 312.48.0-8.e.2.11 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.48.0-312.u.1.33 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.48.0-312.u.1.46 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.48.0-312.y.1.29 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.48.0-312.y.1.33 | $312$ | $2$ | $2$ | $0$ | $?$ |
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.ca.1.4 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.cd.1.5 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.cz.2.9 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.da.2.10 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.eq.2.10 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.er.2.1 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.ew.1.2 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.ex.1.3 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.ga.2.9 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.gb.2.11 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.gc.1.7 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.gd.1.3 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.hw.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.hx.1.4 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.hy.2.2 | $312$ | $2$ | $2$ | $1$ |
| 312.192.1-312.hz.2.9 | $312$ | $2$ | $2$ | $1$ |
| 312.288.8-312.mz.1.39 | $312$ | $3$ | $3$ | $8$ |
| 312.384.7-312.hd.1.59 | $312$ | $4$ | $4$ | $7$ |