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: | $\begin{bmatrix}11&42\\62&55\end{bmatrix}$, $\begin{bmatrix}13&0\\18&67\end{bmatrix}$, $\begin{bmatrix}29&16\\32&51\end{bmatrix}$, $\begin{bmatrix}73&20\\42&65\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 84.96.3.p.1 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
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.1.3 | $28$ | $8$ | $8$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.96.0-12.a.2.15 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 84.96.0-12.a.2.16 | $84$ | $2$ | $2$ | $0$ | $?$ |
| 84.96.1-84.b.1.3 | $84$ | $2$ | $2$ | $1$ | $?$ |
| 84.96.1-84.b.1.8 | $84$ | $2$ | $2$ | $1$ | $?$ |
| 84.96.2-84.a.2.2 | $84$ | $2$ | $2$ | $2$ | $?$ |
| 84.96.2-84.a.2.8 | $84$ | $2$ | $2$ | $2$ | $?$ |
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.g.1.1 | $84$ | $2$ | $2$ | $5$ |
| 84.384.5-84.h.3.2 | $84$ | $2$ | $2$ | $5$ |
| 84.384.5-84.h.4.2 | $84$ | $2$ | $2$ | $5$ |
| 84.384.5-84.m.1.5 | $84$ | $2$ | $2$ | $5$ |
| 84.384.5-84.m.2.5 | $84$ | $2$ | $2$ | $5$ |
| 84.384.5-84.o.1.5 | $84$ | $2$ | $2$ | $5$ |
| 84.384.5-84.o.4.5 | $84$ | $2$ | $2$ | $5$ |
| 168.384.5-168.iw.3.13 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.iw.4.13 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.jd.3.13 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.jd.4.13 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.kh.3.13 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.kh.4.13 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.kv.3.13 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.kv.4.13 | $168$ | $2$ | $2$ | $5$ |