Invariants
| Level: | $84$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $6^{4}\cdot12^{4}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 3$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12D3 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}5&30\\30&23\end{bmatrix}$, $\begin{bmatrix}17&60\\18&53\end{bmatrix}$, $\begin{bmatrix}61&19\\78&11\end{bmatrix}$, $\begin{bmatrix}77&75\\48&65\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 84.72.3.ma.1 for the level structure with $-I$) |
| 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 |
|---|---|---|---|---|---|
| 12.72.0-6.a.1.6 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 42.72.0-6.a.1.1 | $42$ | $2$ | $2$ | $0$ | $0$ |
| 84.48.1-84.o.1.3 | $84$ | $3$ | $3$ | $1$ | $?$ |
| 84.48.1-84.o.1.13 | $84$ | $3$ | $3$ | $1$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.