Invariants
| Level: | $312$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $6^{4}\cdot12^{4}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 3$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12D3 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}19&60\\114&19\end{bmatrix}$, $\begin{bmatrix}31&90\\258&271\end{bmatrix}$, $\begin{bmatrix}233&96\\162&299\end{bmatrix}$, $\begin{bmatrix}257&65\\12&139\end{bmatrix}$, $\begin{bmatrix}307&219\\42&55\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.72.3.cwa.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $56$ |
| Cyclic 312-torsion field degree: | $5376$ |
| Full 312-torsion field degree: | $13418496$ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.72.0-6.a.1.4 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 312.48.1-312.hl.1.6 | $312$ | $3$ | $3$ | $1$ | $?$ |
| 312.48.1-312.hl.1.20 | $312$ | $3$ | $3$ | $1$ | $?$ |
| 312.72.0-6.a.1.4 | $312$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 312.288.5-312.p.1.12 | $312$ | $2$ | $2$ | $5$ |
| 312.288.5-312.dt.1.4 | $312$ | $2$ | $2$ | $5$ |
| 312.288.5-312.ho.1.1 | $312$ | $2$ | $2$ | $5$ |
| 312.288.5-312.hs.1.7 | $312$ | $2$ | $2$ | $5$ |
| 312.288.5-312.qp.1.12 | $312$ | $2$ | $2$ | $5$ |
| 312.288.5-312.qt.1.8 | $312$ | $2$ | $2$ | $5$ |
| 312.288.5-312.rk.1.12 | $312$ | $2$ | $2$ | $5$ |
| 312.288.5-312.ro.1.3 | $312$ | $2$ | $2$ | $5$ |
| 312.288.5-312.bfz.1.6 | $312$ | $2$ | $2$ | $5$ |
| 312.288.5-312.bgc.1.6 | $312$ | $2$ | $2$ | $5$ |
| 312.288.5-312.bgg.1.4 | $312$ | $2$ | $2$ | $5$ |
| 312.288.5-312.bgj.1.4 | $312$ | $2$ | $2$ | $5$ |
| 312.288.5-312.bik.1.6 | $312$ | $2$ | $2$ | $5$ |
| 312.288.5-312.bin.1.1 | $312$ | $2$ | $2$ | $5$ |
| 312.288.5-312.biy.1.6 | $312$ | $2$ | $2$ | $5$ |
| 312.288.5-312.bjk.1.5 | $312$ | $2$ | $2$ | $5$ |