Invariants
| Level: | $312$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $4^{6}\cdot12^{6}$ | Cusp orbits | $1^{2}\cdot2^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 3$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12L3 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}31&230\\228&83\end{bmatrix}$, $\begin{bmatrix}33&32\\88&77\end{bmatrix}$, $\begin{bmatrix}141&92\\190&299\end{bmatrix}$, $\begin{bmatrix}247&14\\18&215\end{bmatrix}$, $\begin{bmatrix}303&242\\70&191\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.96.3.fq.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $56$ |
| Cyclic 312-torsion field degree: | $2688$ |
| Full 312-torsion field degree: | $10063872$ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.96.0-12.a.2.15 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 312.96.0-12.a.2.7 | $312$ | $2$ | $2$ | $0$ | $?$ |
| 312.96.1-312.di.1.21 | $312$ | $2$ | $2$ | $1$ | $?$ |
| 312.96.1-312.di.1.30 | $312$ | $2$ | $2$ | $1$ | $?$ |
| 312.96.2-312.b.1.2 | $312$ | $2$ | $2$ | $2$ | $?$ |
| 312.96.2-312.b.1.24 | $312$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 312.384.5-312.je.2.2 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.jg.2.6 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.jg.3.4 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.jo.1.3 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.jo.3.11 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.jr.1.3 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.jr.4.7 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.jv.3.6 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.jv.4.8 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.jx.3.8 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.jx.4.4 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.kq.2.8 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.kq.4.8 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.ks.3.4 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.ks.4.7 | $312$ | $2$ | $2$ | $5$ |