Invariants
| Level: | $84$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $144$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $6^{8}\cdot12^{8}$ | Cusp orbits | $2^{4}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 5$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12B5 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}5&24\\36&11\end{bmatrix}$, $\begin{bmatrix}11&44\\3&31\end{bmatrix}$, $\begin{bmatrix}17&18\\33&5\end{bmatrix}$, $\begin{bmatrix}47&6\\51&17\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 84.288.5-84.do.1.1, 84.288.5-84.do.1.2, 84.288.5-84.do.1.3, 84.288.5-84.do.1.4, 168.288.5-84.do.1.1, 168.288.5-84.do.1.2, 168.288.5-84.do.1.3, 168.288.5-84.do.1.4 |
| Cyclic 84-isogeny field degree: | $16$ |
| Cyclic 84-torsion field degree: | $384$ |
| Full 84-torsion field degree: | $64512$ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.72.3.cf.1 | $12$ | $2$ | $2$ | $3$ | $0$ |
| 84.48.1.bh.1 | $84$ | $3$ | $3$ | $1$ | $?$ |
| 84.72.1.j.1 | $84$ | $2$ | $2$ | $1$ | $?$ |
| 84.72.1.r.1 | $84$ | $2$ | $2$ | $1$ | $?$ |
| 84.72.1.bs.1 | $84$ | $2$ | $2$ | $1$ | $?$ |
| 84.72.3.jr.1 | $84$ | $2$ | $2$ | $3$ | $?$ |
| 84.72.3.ly.1 | $84$ | $2$ | $2$ | $3$ | $?$ |
| 84.72.3.ox.1 | $84$ | $2$ | $2$ | $3$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 252.432.21.pf.1 | $252$ | $3$ | $3$ | $21$ |
| 252.432.21.pq.1 | $252$ | $3$ | $3$ | $21$ |
| 252.432.21.pv.1 | $252$ | $3$ | $3$ | $21$ |
| 252.432.21.qe.1 | $252$ | $3$ | $3$ | $21$ |
| 252.432.21.qn.1 | $252$ | $3$ | $3$ | $21$ |
| 252.432.21.rj.1 | $252$ | $3$ | $3$ | $21$ |