Invariants
| Level: | $312$ | $\SL_2$-level: | $24$ | Newform level: | $1$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $7 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $6^{4}\cdot12^{6}\cdot24^{2}$ | Cusp orbits | $2^{4}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 12$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 7$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24W7 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}7&186\\248&167\end{bmatrix}$, $\begin{bmatrix}35&136\\4&55\end{bmatrix}$, $\begin{bmatrix}87&118\\184&215\end{bmatrix}$, $\begin{bmatrix}103&308\\36&263\end{bmatrix}$, $\begin{bmatrix}197&200\\200&223\end{bmatrix}$, $\begin{bmatrix}277&68\\80&161\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.144.7.yw.2 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $112$ |
| Cyclic 312-torsion field degree: | $10752$ |
| Full 312-torsion field degree: | $6709248$ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=19$, 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 |
|---|---|---|---|---|---|
| 8.48.0-8.e.1.15 | $8$ | $6$ | $6$ | $0$ | $0$ |
| 39.6.0.a.1 | $39$ | $48$ | $24$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.144.4-24.z.2.47 | $24$ | $2$ | $2$ | $4$ | $0$ |
| 312.144.3-156.k.1.5 | $312$ | $2$ | $2$ | $3$ | $?$ |
| 312.144.3-156.k.1.30 | $312$ | $2$ | $2$ | $3$ | $?$ |
| 312.144.4-24.z.2.45 | $312$ | $2$ | $2$ | $4$ | $?$ |
| 312.144.4-312.bk.1.29 | $312$ | $2$ | $2$ | $4$ | $?$ |
| 312.144.4-312.bk.1.45 | $312$ | $2$ | $2$ | $4$ | $?$ |