Invariants
| Level: | $312$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $4^{6}\cdot12^{6}$ | Cusp orbits | $2^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12K3 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}33&182\\254&171\end{bmatrix}$, $\begin{bmatrix}73&148\\288&161\end{bmatrix}$, $\begin{bmatrix}117&4\\152&95\end{bmatrix}$, $\begin{bmatrix}125&74\\58&69\end{bmatrix}$, $\begin{bmatrix}255&184\\116&275\end{bmatrix}$, $\begin{bmatrix}291&140\\224&99\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.96.3.dq.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 real points and no $\Q_p$ points for $p=23$, and therefore no 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 |
|---|---|---|---|---|---|
| 3.8.0-3.a.1.1 | $3$ | $24$ | $24$ | $0$ | $0$ |
| 104.24.0.c.1 | $104$ | $8$ | $4$ | $0$ | $?$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.96.1-12.a.1.12 | $12$ | $2$ | $2$ | $1$ | $0$ |
| 312.96.1-12.a.1.10 | $312$ | $2$ | $2$ | $1$ | $?$ |
| 312.96.1-312.dh.1.8 | $312$ | $2$ | $2$ | $1$ | $?$ |
| 312.96.1-312.dh.1.9 | $312$ | $2$ | $2$ | $1$ | $?$ |
| 312.96.2-312.s.1.16 | $312$ | $2$ | $2$ | $2$ | $?$ |
| 312.96.2-312.s.1.33 | $312$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 312.384.5-312.is.1.10 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.is.2.2 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.iv.1.14 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.iv.2.10 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.of.1.9 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.of.2.4 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.oh.1.13 | $312$ | $2$ | $2$ | $5$ |
| 312.384.5-312.oh.2.12 | $312$ | $2$ | $2$ | $5$ |
| 312.384.9-312.bi.1.12 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.bl.1.16 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.dt.1.12 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.dv.1.16 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.hj.1.19 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.hj.2.16 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.hl.1.17 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.hl.2.12 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.lj.1.20 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.lj.2.15 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.ll.1.18 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.ll.2.11 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.nn.1.12 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.np.1.16 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.oc.1.12 | $312$ | $2$ | $2$ | $9$ |
| 312.384.9-312.of.1.16 | $312$ | $2$ | $2$ | $9$ |