Invariants
| Level: | $312$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 48$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12P1 |
Level structure
| $\GL_2(\Z/312\Z)$-generators: | $\begin{bmatrix}61&182\\252&53\end{bmatrix}$, $\begin{bmatrix}117&34\\298&159\end{bmatrix}$, $\begin{bmatrix}125&124\\32&81\end{bmatrix}$, $\begin{bmatrix}199&296\\258&209\end{bmatrix}$, $\begin{bmatrix}243&239\\256&53\end{bmatrix}$, $\begin{bmatrix}301&86\\72&185\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 312.48.1.bzq.1 for the level structure with $-I$) |
| Cyclic 312-isogeny field degree: | $56$ |
| Cyclic 312-torsion field degree: | $5376$ |
| Full 312-torsion field degree: | $20127744$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | not computed |
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
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 3.8.0-3.a.1.1 | $3$ | $12$ | $12$ | $0$ | $0$ | full Jacobian |
| 104.12.0.bk.1 | $104$ | $8$ | $4$ | $0$ | $?$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.1-12.l.1.10 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 312.48.0-312.fm.1.5 | $312$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 312.48.0-312.fm.1.26 | $312$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 312.48.0-312.fo.1.2 | $312$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 312.48.0-312.fo.1.29 | $312$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 312.48.1-12.l.1.10 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 312.192.3-312.vq.1.26 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.vr.1.26 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.vy.1.27 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.vz.1.27 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.wi.1.31 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.wj.1.29 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.wk.1.31 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.wl.1.47 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.wm.1.26 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.wn.1.30 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.wo.1.32 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.wp.1.30 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ws.1.27 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.wt.1.27 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ww.1.26 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.wx.1.26 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.5-312.bz.1.30 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.cb.1.30 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.gf.1.30 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.gh.1.30 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.ph.1.30 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.pj.1.30 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.qn.1.30 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.qp.1.30 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.bpo.1.1 | $312$ | $3$ | $3$ | $5$ | $?$ | not computed |