Invariants
| Level: | $84$ | $\SL_2$-level: | $28$ | Newform level: | $1$ | ||
| Index: | $168$ | $\PSL_2$-index: | $168$ | ||||
| Genus: | $11 = 1 + \frac{ 168 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $2$ are rational) | Cusp widths | $14^{4}\cdot28^{4}$ | Cusp orbits | $1^{2}\cdot3^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 11$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 11$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 28A11 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}44&33\\35&26\end{bmatrix}$, $\begin{bmatrix}61&56\\68&9\end{bmatrix}$, $\begin{bmatrix}67&54\\72&17\end{bmatrix}$, $\begin{bmatrix}73&64\\34&81\end{bmatrix}$, $\begin{bmatrix}78&55\\29&42\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 84-isogeny field degree: | $16$ |
| Cyclic 84-torsion field degree: | $384$ |
| Full 84-torsion field degree: | $55296$ |
Rational points
This modular curve has 2 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 |
|---|---|---|---|---|---|
| 14.84.3.a.1 | $14$ | $2$ | $2$ | $3$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 84.336.21.g.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.n.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.bc.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.bf.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.cn.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.cq.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.cr.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.cu.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.di.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.dk.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.dm.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.do.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.dy.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.ea.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.ec.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.ee.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.gd.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.gf.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.gl.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.gn.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.ha.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.hd.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.hi.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.hl.1 | $84$ | $2$ | $2$ | $21$ |
| 168.336.21.n.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.bo.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.dx.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.eg.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.id.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.im.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.ip.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.iy.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.ko.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.ku.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.la.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.lg.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.mk.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.mq.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.mw.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.nc.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.sx.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.td.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.tv.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.ub.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.wb.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.wk.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.wz.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.xi.1 | $168$ | $2$ | $2$ | $21$ |