Invariants
| Level: | $312$ | $\SL_2$-level: | $78$ | Newform level: | $1$ | ||
| Index: | $224$ | $\PSL_2$-index: | $112$ | ||||
| Genus: | $7 = 1 + \frac{ 112 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $2\cdot6\cdot26\cdot78$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $4$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 7$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 7$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 78C7 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}51&308\\113&207\end{bmatrix}$, $\begin{bmatrix}117&233\\2&231\end{bmatrix}$, $\begin{bmatrix}181&272\\40&231\end{bmatrix}$, $\begin{bmatrix}279&193\\247&108\end{bmatrix}$, $\begin{bmatrix}303&154\\103&159\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.112.7.c.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $12$ |
| Cyclic 312-torsion field degree: | $1152$ |
| Full 312-torsion field degree: | $8626176$ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal 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 |
|---|---|---|---|---|---|
| $X_0(13)$ | $13$ | $16$ | $8$ | $0$ | $0$ |
| 24.16.0-24.c.1.6 | $24$ | $14$ | $14$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.16.0-24.c.1.6 | $24$ | $14$ | $14$ | $0$ | $0$ |
| 156.112.3-39.a.1.12 | $156$ | $2$ | $2$ | $3$ | $?$ |
| 312.112.3-39.a.1.1 | $312$ | $2$ | $2$ | $3$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 312.448.13-312.bl.1.7 | $312$ | $2$ | $2$ | $13$ |
| 312.448.13-312.bl.2.5 | $312$ | $2$ | $2$ | $13$ |
| 312.448.13-312.bm.1.7 | $312$ | $2$ | $2$ | $13$ |
| 312.448.13-312.bm.2.5 | $312$ | $2$ | $2$ | $13$ |
| 312.448.13-312.bo.1.4 | $312$ | $2$ | $2$ | $13$ |
| 312.448.13-312.bo.2.2 | $312$ | $2$ | $2$ | $13$ |
| 312.448.13-312.bp.1.7 | $312$ | $2$ | $2$ | $13$ |
| 312.448.13-312.bp.2.3 | $312$ | $2$ | $2$ | $13$ |
| 312.448.13-312.cj.1.4 | $312$ | $2$ | $2$ | $13$ |
| 312.448.13-312.cj.2.3 | $312$ | $2$ | $2$ | $13$ |
| 312.448.13-312.ck.1.7 | $312$ | $2$ | $2$ | $13$ |
| 312.448.13-312.ck.2.5 | $312$ | $2$ | $2$ | $13$ |
| 312.448.13-312.cm.1.4 | $312$ | $2$ | $2$ | $13$ |
| 312.448.13-312.cm.2.2 | $312$ | $2$ | $2$ | $13$ |
| 312.448.13-312.cn.1.7 | $312$ | $2$ | $2$ | $13$ |
| 312.448.13-312.cn.2.3 | $312$ | $2$ | $2$ | $13$ |