Invariants
| Level: | $84$ | $\SL_2$-level: | $42$ | Newform level: | $1$ | ||
| Index: | $256$ | $\PSL_2$-index: | $128$ | ||||
| Genus: | $5 = 1 + \frac{ 128 }{12} - \frac{ 0 }{4} - \frac{ 8 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot6^{2}\cdot14^{2}\cdot42^{2}$ | Cusp orbits | $1^{2}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $8$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 5$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 5$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 42I5 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}6&67\\7&30\end{bmatrix}$, $\begin{bmatrix}8&7\\71&72\end{bmatrix}$, $\begin{bmatrix}31&15\\48&1\end{bmatrix}$, $\begin{bmatrix}45&56\\23&75\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 84.128.5.c.3 for the level structure with $-I$) |
| Cyclic 84-isogeny field degree: | $6$ |
| Cyclic 84-torsion field degree: | $144$ |
| Full 84-torsion field degree: | $36288$ |
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 |
|---|---|---|---|---|---|
| 4.2.0.a.1 | $4$ | $128$ | $64$ | $0$ | $0$ |
| 21.128.1-21.a.2.4 | $21$ | $2$ | $2$ | $1$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 21.128.1-21.a.2.4 | $21$ | $2$ | $2$ | $1$ | $0$ |
| 84.128.1-21.a.2.6 | $84$ | $2$ | $2$ | $1$ | $?$ |
| 84.128.3-84.a.1.13 | $84$ | $2$ | $2$ | $3$ | $?$ |
| 84.128.3-84.a.1.21 | $84$ | $2$ | $2$ | $3$ | $?$ |
| 84.128.3-84.g.3.8 | $84$ | $2$ | $2$ | $3$ | $?$ |
| 84.128.3-84.g.3.10 | $84$ | $2$ | $2$ | $3$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 84.512.13-84.d.3.7 | $84$ | $2$ | $2$ | $13$ |
| 84.512.13-84.h.1.8 | $84$ | $2$ | $2$ | $13$ |
| 84.512.13-84.l.4.3 | $84$ | $2$ | $2$ | $13$ |
| 84.512.13-84.p.2.6 | $84$ | $2$ | $2$ | $13$ |
| 168.512.13-168.j.3.16 | $168$ | $2$ | $2$ | $13$ |
| 168.512.13-168.v.1.16 | $168$ | $2$ | $2$ | $13$ |
| 168.512.13-168.bh.4.8 | $168$ | $2$ | $2$ | $13$ |
| 168.512.13-168.bt.2.12 | $168$ | $2$ | $2$ | $13$ |