Invariants
| Level: | $84$ | $\SL_2$-level: | $84$ | Newform level: | $1$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $11 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $4$ are rational) | Cusp widths | $1^{2}\cdot3^{2}\cdot4\cdot7^{2}\cdot12\cdot21^{2}\cdot28\cdot84$ | Cusp orbits | $1^{4}\cdot2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 11$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 11$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 84I11 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}8&25\\77&12\end{bmatrix}$, $\begin{bmatrix}8&43\\5&60\end{bmatrix}$, $\begin{bmatrix}13&0\\70&41\end{bmatrix}$, $\begin{bmatrix}35&24\\66&77\end{bmatrix}$, $\begin{bmatrix}43&18\\16&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 84.192.11.cm.1 for the level structure with $-I$) |
| Cyclic 84-isogeny field degree: | $2$ |
| Cyclic 84-torsion field degree: | $48$ |
| Full 84-torsion field degree: | $24192$ |
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 |
|---|---|---|---|---|---|
| 42.192.5-42.a.1.3 | $42$ | $2$ | $2$ | $5$ | $0$ |
| 84.48.0-84.q.1.3 | $84$ | $8$ | $8$ | $0$ | $?$ |
| 84.192.5-42.a.1.47 | $84$ | $2$ | $2$ | $5$ | $?$ |