Invariants
| Level: | $84$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $6^{4}\cdot12^{4}$ | Cusp orbits | $2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12G3 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}13&34\\46&9\end{bmatrix}$, $\begin{bmatrix}23&4\\30&37\end{bmatrix}$, $\begin{bmatrix}27&28\\34&17\end{bmatrix}$, $\begin{bmatrix}39&40\\56&69\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 84.72.3.o.1 for the level structure with $-I$) |
| Cyclic 84-isogeny field degree: | $64$ |
| Cyclic 84-torsion field degree: | $768$ |
| Full 84-torsion field degree: | $64512$ |
Rational points
This modular curve has no $\Q_p$ points for $p=19$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.72.2-12.d.1.3 | $12$ | $2$ | $2$ | $2$ | $0$ |
| 84.72.1-42.a.1.2 | $84$ | $2$ | $2$ | $1$ | $?$ |
| 84.72.1-42.a.1.3 | $84$ | $2$ | $2$ | $1$ | $?$ |
| 84.72.2-12.d.1.4 | $84$ | $2$ | $2$ | $2$ | $?$ |
| 84.72.2-84.e.1.5 | $84$ | $2$ | $2$ | $2$ | $?$ |
| 84.72.2-84.e.1.9 | $84$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 84.288.7-84.k.1.1 | $84$ | $2$ | $2$ | $7$ |
| 84.288.7-84.k.1.4 | $84$ | $2$ | $2$ | $7$ |
| 84.288.7-84.l.1.1 | $84$ | $2$ | $2$ | $7$ |
| 84.288.7-84.l.1.8 | $84$ | $2$ | $2$ | $7$ |
| 84.288.7-84.de.1.2 | $84$ | $2$ | $2$ | $7$ |
| 84.288.7-84.de.1.7 | $84$ | $2$ | $2$ | $7$ |
| 84.288.7-84.df.1.3 | $84$ | $2$ | $2$ | $7$ |
| 84.288.7-84.df.1.6 | $84$ | $2$ | $2$ | $7$ |
| 168.288.7-168.bs.1.2 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.bs.1.15 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.by.1.2 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.by.1.15 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.sn.1.2 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.sn.1.15 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.su.1.2 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.su.1.15 | $168$ | $2$ | $2$ | $7$ |
| 252.432.15-252.bf.1.6 | $252$ | $3$ | $3$ | $15$ |