Invariants
| Level: | $84$ | $\SL_2$-level: | $4$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4E0 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}5&78\\38&83\end{bmatrix}$, $\begin{bmatrix}17&56\\64&75\end{bmatrix}$, $\begin{bmatrix}59&36\\6&77\end{bmatrix}$, $\begin{bmatrix}83&8\\8&57\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 84.12.0.b.1 for the level structure with $-I$) |
| Cyclic 84-isogeny field degree: | $64$ |
| Cyclic 84-torsion field degree: | $1536$ |
| Full 84-torsion field degree: | $387072$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.12.0-2.a.1.2 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 28.12.0-2.a.1.1 | $28$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 84.48.0-84.b.1.2 | $84$ | $2$ | $2$ | $0$ |
| 84.48.0-84.c.1.2 | $84$ | $2$ | $2$ | $0$ |
| 84.48.0-84.c.1.3 | $84$ | $2$ | $2$ | $0$ |
| 84.48.0-84.e.1.1 | $84$ | $2$ | $2$ | $0$ |
| 84.48.0-84.e.1.4 | $84$ | $2$ | $2$ | $0$ |
| 84.48.0-84.f.1.2 | $84$ | $2$ | $2$ | $0$ |
| 84.48.0-84.f.1.3 | $84$ | $2$ | $2$ | $0$ |
| 84.72.2-84.d.1.4 | $84$ | $3$ | $3$ | $2$ |
| 84.96.1-84.d.1.6 | $84$ | $4$ | $4$ | $1$ |
| 84.192.5-84.d.1.17 | $84$ | $8$ | $8$ | $5$ |
| 84.504.16-84.d.1.13 | $84$ | $21$ | $21$ | $16$ |
| 168.48.0-168.d.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.d.1.11 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.g.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.g.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.m.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.m.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.p.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.p.1.9 | $168$ | $2$ | $2$ | $0$ |