Invariants
| Level: | $84$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $4^{6}\cdot12^{6}$ | Cusp orbits | $1^{2}\cdot2^{5}$ | ||
| 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: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12L3 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}11&18\\50&31\end{bmatrix}$, $\begin{bmatrix}21&44\\26&63\end{bmatrix}$, $\begin{bmatrix}39&80\\2&27\end{bmatrix}$, $\begin{bmatrix}75&58\\28&39\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 84.96.3.r.2 for the level structure with $-I$) |
| Cyclic 84-isogeny field degree: | $16$ |
| Cyclic 84-torsion field degree: | $192$ |
| Full 84-torsion field degree: | $48384$ |
Rational points
This modular curve has 2 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 |
|---|---|---|---|---|---|
| 12.96.0-12.a.2.9 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 84.96.0-12.a.2.12 | $84$ | $2$ | $2$ | $0$ | $?$ |
| 84.96.1-84.d.1.5 | $84$ | $2$ | $2$ | $1$ | $?$ |
| 84.96.1-84.d.1.9 | $84$ | $2$ | $2$ | $1$ | $?$ |
| 84.96.2-84.a.1.5 | $84$ | $2$ | $2$ | $2$ | $?$ |
| 84.96.2-84.a.1.11 | $84$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 84.384.5-84.j.2.3 | $84$ | $2$ | $2$ | $5$ |
| 84.384.5-84.j.2.7 | $84$ | $2$ | $2$ | $5$ |
| 84.384.5-84.k.2.4 | $84$ | $2$ | $2$ | $5$ |
| 84.384.5-84.k.3.8 | $84$ | $2$ | $2$ | $5$ |
| 84.384.5-84.m.3.4 | $84$ | $2$ | $2$ | $5$ |
| 84.384.5-84.m.4.7 | $84$ | $2$ | $2$ | $5$ |
| 84.384.5-84.n.2.4 | $84$ | $2$ | $2$ | $5$ |
| 84.384.5-84.n.4.8 | $84$ | $2$ | $2$ | $5$ |
| 168.384.5-168.jn.1.5 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.jn.3.7 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.ju.1.7 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.ju.3.8 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.ki.1.5 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.ki.3.7 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.kp.1.7 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.kp.3.8 | $168$ | $2$ | $2$ | $5$ |