Invariants
| Level: | $312$ | $\SL_2$-level: | $24$ | Newform level: | $1$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $6^{4}\cdot24^{2}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24D4 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}5&70\\148&97\end{bmatrix}$, $\begin{bmatrix}53&41\\68&163\end{bmatrix}$, $\begin{bmatrix}85&72\\8&221\end{bmatrix}$, $\begin{bmatrix}99&112\\200&159\end{bmatrix}$, $\begin{bmatrix}139&180\\140&59\end{bmatrix}$, $\begin{bmatrix}145&292\\64&177\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.72.4.hw.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $112$ |
| Cyclic 312-torsion field degree: | $10752$ |
| Full 312-torsion field degree: | $13418496$ |
Rational points
This modular curve has no real points, 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 |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $48$ | $24$ | $0$ | $0$ |
| 104.48.0-104.be.1.4 | $104$ | $3$ | $3$ | $0$ | $?$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.72.2-12.p.1.14 | $24$ | $2$ | $2$ | $2$ | $0$ |
| 104.48.0-104.be.1.4 | $104$ | $3$ | $3$ | $0$ | $?$ |
| 312.72.2-12.p.1.7 | $312$ | $2$ | $2$ | $2$ | $?$ |
| 312.72.2-312.di.1.3 | $312$ | $2$ | $2$ | $2$ | $?$ |
| 312.72.2-312.di.1.7 | $312$ | $2$ | $2$ | $2$ | $?$ |
| 312.72.2-312.di.1.58 | $312$ | $2$ | $2$ | $2$ | $?$ |
| 312.72.2-312.di.1.62 | $312$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.