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 | $2^{3}\cdot6^{3}\cdot14^{3}\cdot42^{3}$ | 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: | 42G11 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}6&59\\7&74\end{bmatrix}$, $\begin{bmatrix}9&34\\70&81\end{bmatrix}$, $\begin{bmatrix}32&67\\63&22\end{bmatrix}$, $\begin{bmatrix}33&26\\70&65\end{bmatrix}$, $\begin{bmatrix}60&11\\35&66\end{bmatrix}$, $\begin{bmatrix}79&38\\42&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 84.192.11.z.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
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(7)$ | $7$ | $48$ | $24$ | $0$ | $0$ |
| 12.48.0-12.d.1.10 | $12$ | $8$ | $8$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.48.0-12.d.1.10 | $12$ | $8$ | $8$ | $0$ | $0$ |
| 84.192.5-42.a.1.41 | $84$ | $2$ | $2$ | $5$ | $?$ |
| 84.192.5-42.a.1.47 | $84$ | $2$ | $2$ | $5$ | $?$ |