Invariants
| Level: | $84$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $144$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $4$ are rational) | Cusp widths | $6^{8}\cdot12^{8}$ | Cusp orbits | $1^{4}\cdot2^{4}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 5$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 5$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12B5 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}31&54\\54&25\end{bmatrix}$, $\begin{bmatrix}31&56\\42&5\end{bmatrix}$, $\begin{bmatrix}53&42\\60&5\end{bmatrix}$, $\begin{bmatrix}53&54\\12&59\end{bmatrix}$, $\begin{bmatrix}67&74\\78&53\end{bmatrix}$, $\begin{bmatrix}79&38\\48&83\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 84.288.5-84.f.1.1, 84.288.5-84.f.1.2, 84.288.5-84.f.1.3, 84.288.5-84.f.1.4, 84.288.5-84.f.1.5, 84.288.5-84.f.1.6, 84.288.5-84.f.1.7, 84.288.5-84.f.1.8, 84.288.5-84.f.1.9, 84.288.5-84.f.1.10, 84.288.5-84.f.1.11, 84.288.5-84.f.1.12, 84.288.5-84.f.1.13, 84.288.5-84.f.1.14, 168.288.5-84.f.1.1, 168.288.5-84.f.1.2, 168.288.5-84.f.1.3, 168.288.5-84.f.1.4, 168.288.5-84.f.1.5, 168.288.5-84.f.1.6, 168.288.5-84.f.1.7, 168.288.5-84.f.1.8, 168.288.5-84.f.1.9, 168.288.5-84.f.1.10, 168.288.5-84.f.1.11, 168.288.5-84.f.1.12, 168.288.5-84.f.1.13, 168.288.5-84.f.1.14 |
| Cyclic 84-isogeny field degree: | $16$ |
| Cyclic 84-torsion field degree: | $384$ |
| Full 84-torsion field degree: | $64512$ |
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 |
|---|---|---|---|---|---|
| $X_{\mathrm{sp}}(6)$ | $6$ | $2$ | $2$ | $1$ | $0$ |
| 84.48.1.c.1 | $84$ | $3$ | $3$ | $1$ | $?$ |
| 84.72.1.r.1 | $84$ | $2$ | $2$ | $1$ | $?$ |
| 84.72.1.cd.1 | $84$ | $2$ | $2$ | $1$ | $?$ |
| 84.72.3.bc.1 | $84$ | $2$ | $2$ | $3$ | $?$ |
| 84.72.3.bx.1 | $84$ | $2$ | $2$ | $3$ | $?$ |
| 84.72.3.jw.1 | $84$ | $2$ | $2$ | $3$ | $?$ |
| 84.72.3.ra.1 | $84$ | $2$ | $2$ | $3$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 84.288.9.f.1 | $84$ | $2$ | $2$ | $9$ |
| 84.288.9.f.2 | $84$ | $2$ | $2$ | $9$ |
| 84.288.13.r.1 | $84$ | $2$ | $2$ | $13$ |
| 84.288.13.s.1 | $84$ | $2$ | $2$ | $13$ |
| 84.288.13.x.1 | $84$ | $2$ | $2$ | $13$ |
| 84.288.13.z.1 | $84$ | $2$ | $2$ | $13$ |
| 84.288.13.bh.1 | $84$ | $2$ | $2$ | $13$ |
| 84.288.13.bi.1 | $84$ | $2$ | $2$ | $13$ |
| 168.288.9.p.1 | $168$ | $2$ | $2$ | $9$ |
| 168.288.9.p.2 | $168$ | $2$ | $2$ | $9$ |
| 168.288.13.ey.1 | $168$ | $2$ | $2$ | $13$ |
| 168.288.13.fe.1 | $168$ | $2$ | $2$ | $13$ |
| 168.288.13.hl.1 | $168$ | $2$ | $2$ | $13$ |
| 168.288.13.hr.1 | $168$ | $2$ | $2$ | $13$ |
| 168.288.13.ji.1 | $168$ | $2$ | $2$ | $13$ |
| 168.288.13.jp.1 | $168$ | $2$ | $2$ | $13$ |
| 252.432.21.d.1 | $252$ | $3$ | $3$ | $21$ |
| 252.432.21.g.1 | $252$ | $3$ | $3$ | $21$ |
| 252.432.21.r.1 | $252$ | $3$ | $3$ | $21$ |
| 252.432.21.z.1 | $252$ | $3$ | $3$ | $21$ |
| 252.432.21.bl.1 | $252$ | $3$ | $3$ | $21$ |
| 252.432.21.bp.1 | $252$ | $3$ | $3$ | $21$ |