Invariants
| Level: | $312$ | $\SL_2$-level: | $24$ | Newform level: | $1$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $5 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $4^{2}\cdot8^{2}\cdot12^{2}\cdot24^{2}$ | Cusp orbits | $2^{4}$ | ||
| 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: | 24I5 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}5&160\\176&297\end{bmatrix}$, $\begin{bmatrix}91&278\\154&69\end{bmatrix}$, $\begin{bmatrix}149&156\\18&275\end{bmatrix}$, $\begin{bmatrix}185&286\\296&297\end{bmatrix}$, $\begin{bmatrix}205&131\\240&155\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.96.5.iy.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $56$ |
| Cyclic 312-torsion field degree: | $5376$ |
| Full 312-torsion field degree: | $10063872$ |
Rational points
This modular curve has no $\Q_p$ points for $p=29$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.96.1-12.o.1.3 | $12$ | $2$ | $2$ | $1$ | $0$ |
| 312.96.1-12.o.1.11 | $312$ | $2$ | $2$ | $1$ | $?$ |
| 312.96.3-312.cf.1.24 | $312$ | $2$ | $2$ | $3$ | $?$ |
| 312.96.3-312.cf.1.31 | $312$ | $2$ | $2$ | $3$ | $?$ |
| 312.96.3-312.cm.1.24 | $312$ | $2$ | $2$ | $3$ | $?$ |
| 312.96.3-312.cm.1.31 | $312$ | $2$ | $2$ | $3$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 312.384.9-312.cml.1.7 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.cmm.1.5 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.cmn.1.15 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.cmo.1.4 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.cnr.1.5 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.cns.1.7 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.cnt.1.15 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.cnu.1.3 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.coh.1.15 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.coi.1.3 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.coj.1.7 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.cok.1.5 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.cox.1.15 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.coy.1.4 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.coz.1.5 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.cpa.1.7 | $312$ | $2$ | $2$ | $9$ |