Invariants
| Level: | $312$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $3 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 20 }{2}$ | ||||||
| Cusps: | $20$ (of which $2$ are rational) | Cusp widths | $6^{16}\cdot12^{4}$ | Cusp orbits | $1^{2}\cdot2^{5}\cdot4^{2}$ | ||
| 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: | 12N3 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}1&174\\210&13\end{bmatrix}$, $\begin{bmatrix}199&222\\204&277\end{bmatrix}$, $\begin{bmatrix}271&240\\0&187\end{bmatrix}$, $\begin{bmatrix}283&108\\18&31\end{bmatrix}$, $\begin{bmatrix}301&164\\192&101\end{bmatrix}$, $\begin{bmatrix}301&236\\240&119\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.144.3.a.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $56$ |
| Cyclic 312-torsion field degree: | $2688$ |
| Full 312-torsion field degree: | $6709248$ |
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 |
|---|---|---|---|---|---|
| $X_{\mathrm{arith}}(6)$ | $6$ | $2$ | $2$ | $1$ | $0$ |
| 312.96.0-312.o.1.32 | $312$ | $3$ | $3$ | $0$ | $?$ |
| 312.96.0-312.o.2.56 | $312$ | $3$ | $3$ | $0$ | $?$ |
| 312.144.1-6.a.1.3 | $312$ | $2$ | $2$ | $1$ | $?$ |