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: | $\begin{bmatrix}69&284\\68&191\end{bmatrix}$, $\begin{bmatrix}71&61\\124&235\end{bmatrix}$, $\begin{bmatrix}145&160\\276&203\end{bmatrix}$, $\begin{bmatrix}275&241\\280&69\end{bmatrix}$, $\begin{bmatrix}307&112\\16&41\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.72.4.oe.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $112$ |
| Cyclic 312-torsion field degree: | $5376$ |
| Full 312-torsion field degree: | $13418496$ |
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
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.72.2-24.cu.1.27 | $24$ | $2$ | $2$ | $2$ | $0$ |
| 156.72.2-156.x.1.5 | $156$ | $2$ | $2$ | $2$ | $?$ |
| 312.48.0-312.ea.1.10 | $312$ | $3$ | $3$ | $0$ | $?$ |
| 312.72.2-156.x.1.18 | $312$ | $2$ | $2$ | $2$ | $?$ |
| 312.72.2-24.cu.1.16 | $312$ | $2$ | $2$ | $2$ | $?$ |
| 312.72.2-312.di.1.23 | $312$ | $2$ | $2$ | $2$ | $?$ |
| 312.72.2-312.di.1.26 | $312$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 312.288.7-312.dwk.1.3 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.dwm.1.5 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.dxa.1.7 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.dxc.1.5 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.eii.1.19 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.eik.1.10 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.ejc.1.3 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.eje.1.10 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.etg.1.5 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.eti.1.6 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.etw.1.5 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.ety.1.6 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.fea.1.10 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.fec.1.10 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.feq.1.10 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.fes.1.10 | $312$ | $2$ | $2$ | $7$ |