Invariants
| Level: | $312$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $144$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $6^{8}\cdot12^{8}$ | Cusp orbits | $2^{4}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 5$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12B5 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}107&200\\3&199\end{bmatrix}$, $\begin{bmatrix}143&270\\18&263\end{bmatrix}$, $\begin{bmatrix}209&164\\192&103\end{bmatrix}$, $\begin{bmatrix}223&252\\48&217\end{bmatrix}$, $\begin{bmatrix}299&216\\60&53\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 312.288.5-312.bps.1.1, 312.288.5-312.bps.1.2, 312.288.5-312.bps.1.3, 312.288.5-312.bps.1.4, 312.288.5-312.bps.1.5, 312.288.5-312.bps.1.6, 312.288.5-312.bps.1.7, 312.288.5-312.bps.1.8 |
| Cyclic 312-isogeny field degree: | $56$ |
| Cyclic 312-torsion field degree: | $5376$ |
| Full 312-torsion field degree: | $13418496$ |
Rational points
This modular curve has no $\Q_p$ points for $p=7,31$, and therefore no 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 |
|---|---|---|---|---|---|
| $X_{\mathrm{sp}}(3)$ | $3$ | $12$ | $12$ | $0$ | $0$ |
| 104.12.0.bm.1 | $104$ | $12$ | $12$ | $0$ | $?$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.72.3.ug.1 | $24$ | $2$ | $2$ | $3$ | $0$ |
| 156.72.1.n.1 | $156$ | $2$ | $2$ | $1$ | $?$ |
| 312.48.1.bzs.1 | $312$ | $3$ | $3$ | $1$ | $?$ |
| 312.72.1.bf.1 | $312$ | $2$ | $2$ | $1$ | $?$ |
| 312.72.1.ig.1 | $312$ | $2$ | $2$ | $1$ | $?$ |
| 312.72.3.cvh.1 | $312$ | $2$ | $2$ | $3$ | $?$ |
| 312.72.3.cvs.1 | $312$ | $2$ | $2$ | $3$ | $?$ |
| 312.72.3.elg.1 | $312$ | $2$ | $2$ | $3$ | $?$ |