Invariants
| Level: | $312$ | $\SL_2$-level: | $24$ | Newform level: | $1$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $8 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (none of which are rational) | Cusp widths | $6^{4}\cdot12^{2}\cdot24^{4}$ | Cusp orbits | $2^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 14$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 8$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24B8 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}1&126\\212&35\end{bmatrix}$, $\begin{bmatrix}31&182\\96&119\end{bmatrix}$, $\begin{bmatrix}37&152\\200&215\end{bmatrix}$, $\begin{bmatrix}123&196\\104&159\end{bmatrix}$, $\begin{bmatrix}133&124\\100&27\end{bmatrix}$, $\begin{bmatrix}257&62\\164&115\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.144.8.np.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 real points and $\Q_p$ points for $p$ not dividing the level, but no known 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 |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $96$ | $48$ | $0$ | $0$ |
| 104.96.0-104.t.1.9 | $104$ | $3$ | $3$ | $0$ | $?$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.144.4-24.z.2.48 | $24$ | $2$ | $2$ | $4$ | $0$ |
| 104.96.0-104.t.1.9 | $104$ | $3$ | $3$ | $0$ | $?$ |
| 312.144.4-24.z.2.55 | $312$ | $2$ | $2$ | $4$ | $?$ |
| 312.144.4-312.bp.1.48 | $312$ | $2$ | $2$ | $4$ | $?$ |
| 312.144.4-312.bp.1.107 | $312$ | $2$ | $2$ | $4$ | $?$ |
| 312.144.4-312.es.1.43 | $312$ | $2$ | $2$ | $4$ | $?$ |
| 312.144.4-312.es.1.57 | $312$ | $2$ | $2$ | $4$ | $?$ |