Invariants
| Level: | $312$ | $\SL_2$-level: | $12$ | 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 | $12^{6}$ | 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: | 12A4 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}13&200\\38&131\end{bmatrix}$, $\begin{bmatrix}27&50\\98&107\end{bmatrix}$, $\begin{bmatrix}63&310\\16&151\end{bmatrix}$, $\begin{bmatrix}97&168\\102&115\end{bmatrix}$, $\begin{bmatrix}223&242\\14&27\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.72.4.u.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $224$ |
| Cyclic 312-torsion field degree: | $10752$ |
| 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 |
|---|---|---|---|---|---|
| 12.72.2-12.d.1.3 | $12$ | $2$ | $2$ | $2$ | $0$ |
| 312.48.0-312.n.1.15 | $312$ | $3$ | $3$ | $0$ | $?$ |
| 312.72.2-312.a.1.27 | $312$ | $2$ | $2$ | $2$ | $?$ |
| 312.72.2-312.a.1.32 | $312$ | $2$ | $2$ | $2$ | $?$ |
| 312.72.2-12.d.1.3 | $312$ | $2$ | $2$ | $2$ | $?$ |
| 312.72.2-312.d.1.4 | $312$ | $2$ | $2$ | $2$ | $?$ |
| 312.72.2-312.d.1.23 | $312$ | $2$ | $2$ | $2$ | $?$ |
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.hj.1.13 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.hm.1.14 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.ie.1.13 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.ih.1.15 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.ml.1.7 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.mo.1.1 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.ng.1.1 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.nj.1.1 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.su.1.15 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.sx.1.16 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.ti.1.9 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.tl.1.13 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.vz.1.7 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.wc.1.1 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.xb.1.14 | $312$ | $2$ | $2$ | $7$ |
| 312.288.7-312.xe.1.8 | $312$ | $2$ | $2$ | $7$ |