Invariants
| Level: | $312$ | $\SL_2$-level: | $52$ | Newform level: | $1$ | ||
| Index: | $336$ | $\PSL_2$-index: | $168$ | ||||
| Genus: | $11 = 1 + \frac{ 168 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot26^{2}\cdot52^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 11$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 11$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 52D11 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}33&52\\272&209\end{bmatrix}$, $\begin{bmatrix}33&208\\76&59\end{bmatrix}$, $\begin{bmatrix}151&234\\76&41\end{bmatrix}$, $\begin{bmatrix}187&208\\132&203\end{bmatrix}$, $\begin{bmatrix}187&234\\182&121\end{bmatrix}$, $\begin{bmatrix}217&26\\196&133\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 104.168.11.b.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $16$ |
| Cyclic 312-torsion field degree: | $1536$ |
| Full 312-torsion field degree: | $5750784$ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(13)$ | $13$ | $24$ | $12$ | $0$ | $0$ |
| 24.24.0-8.b.1.4 | $24$ | $14$ | $14$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.24.0-8.b.1.4 | $24$ | $14$ | $14$ | $0$ | $0$ |
| 156.168.5-26.a.1.5 | $156$ | $2$ | $2$ | $5$ | $?$ |
| 312.168.5-26.a.1.1 | $312$ | $2$ | $2$ | $5$ | $?$ |