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 $6$ are rational) | Cusp widths | $1^{2}\cdot3^{2}\cdot4\cdot7^{2}\cdot12\cdot21^{2}\cdot28\cdot84$ | Cusp orbits | $1^{6}\cdot2^{3}$ | ||
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: | $6$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 84H11 |
Level structure
$\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}1&2\\66&77\end{bmatrix}$, $\begin{bmatrix}11&0\\8&73\end{bmatrix}$, $\begin{bmatrix}31&14\\40&33\end{bmatrix}$, $\begin{bmatrix}40&3\\51&76\end{bmatrix}$, $\begin{bmatrix}78&5\\67&2\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 84.192.11.cs.2 for the level structure with $-I$) |
Cyclic 84-isogeny field degree: | $2$ |
Cyclic 84-torsion field degree: | $24$ |
Full 84-torsion field degree: | $24192$ |
Rational points
This modular curve has 6 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 |
---|---|---|---|---|---|
84.192.5-42.a.1.6 | $84$ | $2$ | $2$ | $5$ | $?$ |
84.192.5-42.a.1.47 | $84$ | $2$ | $2$ | $5$ | $?$ |