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: | $12$ (none of which are rational) | Cusp widths | $4^{6}\cdot12^{6}$ | Cusp orbits | $2^{6}$ | ||
| 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: | 12L3 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}7&62\\36&23\end{bmatrix}$, $\begin{bmatrix}17&4\\62&51\end{bmatrix}$, $\begin{bmatrix}59&48\\80&37\end{bmatrix}$, $\begin{bmatrix}79&60\\0&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 84.96.3.t.2 for the level structure with $-I$) |
| Cyclic 84-isogeny field degree: | $16$ |
| Cyclic 84-torsion field degree: | $384$ |
| Full 84-torsion field degree: | $48384$ |
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
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 3.8.0-3.a.1.1 | $3$ | $24$ | $24$ | $0$ | $0$ |
| 28.24.0-28.b.1.4 | $28$ | $8$ | $8$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.96.2-12.a.2.8 | $12$ | $2$ | $2$ | $2$ | $0$ |
| 84.96.0-84.a.2.8 | $84$ | $2$ | $2$ | $0$ | $?$ |
| 84.96.0-84.a.2.28 | $84$ | $2$ | $2$ | $0$ | $?$ |
| 84.96.1-84.b.1.8 | $84$ | $2$ | $2$ | $1$ | $?$ |
| 84.96.1-84.b.1.13 | $84$ | $2$ | $2$ | $1$ | $?$ |
| 84.96.2-12.a.2.5 | $84$ | $2$ | $2$ | $2$ | $?$ |
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.q.2.6 | $84$ | $2$ | $2$ | $5$ |
| 84.384.5-84.r.2.6 | $84$ | $2$ | $2$ | $5$ |
| 84.384.5-84.r.4.4 | $84$ | $2$ | $2$ | $5$ |
| 84.384.5-84.w.1.7 | $84$ | $2$ | $2$ | $5$ |
| 84.384.5-84.w.3.6 | $84$ | $2$ | $2$ | $5$ |
| 84.384.5-84.y.1.8 | $84$ | $2$ | $2$ | $5$ |
| 84.384.5-84.y.4.8 | $84$ | $2$ | $2$ | $5$ |
| 168.384.5-168.oi.3.14 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.oi.4.14 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.op.3.14 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.op.4.14 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.pt.3.14 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.pt.4.14 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.qh.3.14 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.qh.4.14 | $168$ | $2$ | $2$ | $5$ |