Invariants
| Level: | $84$ | $\SL_2$-level: | $28$ | Newform level: | $1$ | ||
| Index: | $336$ | $\PSL_2$-index: | $336$ | ||||
| Genus: | $21 = 1 + \frac{ 336 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $14^{8}\cdot28^{8}$ | Cusp orbits | $2^{2}\cdot6^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $5 \le \gamma \le 40$ | ||||||
| $\overline{\Q}$-gonality: | $5 \le \gamma \le 21$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 28E21 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}22&19\\9&34\end{bmatrix}$, $\begin{bmatrix}32&13\\31&66\end{bmatrix}$, $\begin{bmatrix}34&41\\57&40\end{bmatrix}$, $\begin{bmatrix}39&26\\44&31\end{bmatrix}$ |
| 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: | $27648$ |
Rational points
This modular curve has no $\Q_p$ points for $p=5,17,29$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 28.168.11.b.1 | $28$ | $2$ | $2$ | $11$ | $2$ |
| 42.168.7.a.1 | $42$ | $2$ | $2$ | $7$ | $2$ |
| 84.168.9.bg.1 | $84$ | $2$ | $2$ | $9$ | $?$ |