Invariants
| Level: | $84$ | $\SL_2$-level: | $6$ | ||||
| Index: | $8$ | $\PSL_2$-index: | $4$ | ||||
| Genus: | $0 = 1 + \frac{ 4 }{12} - \frac{ 0 }{4} - \frac{ 1 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $1\cdot3$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $1$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 3B0 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}0&11\\65&69\end{bmatrix}$, $\begin{bmatrix}23&45\\62&49\end{bmatrix}$, $\begin{bmatrix}67&51\\52&83\end{bmatrix}$, $\begin{bmatrix}67&80\\81&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 3.4.0.a.1 for the level structure with $-I$) |
| Cyclic 84-isogeny field degree: | $48$ |
| Cyclic 84-torsion field degree: | $1152$ |
| Full 84-torsion field degree: | $1161216$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 78278 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 4 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^3}\cdot\frac{x^{4}(x-18y)^{3}(x+30y)}{y^{3}x^{4}(x-24y)}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 84.16.0-6.a.1.2 | $84$ | $2$ | $2$ | $0$ |
| 84.16.0-12.a.1.2 | $84$ | $2$ | $2$ | $0$ |
| 84.16.0-42.a.1.4 | $84$ | $2$ | $2$ | $0$ |
| 84.16.0-84.a.1.5 | $84$ | $2$ | $2$ | $0$ |
| 84.16.0-6.b.1.1 | $84$ | $2$ | $2$ | $0$ |
| 84.16.0-12.b.1.4 | $84$ | $2$ | $2$ | $0$ |
| 84.16.0-42.b.1.3 | $84$ | $2$ | $2$ | $0$ |
| 84.16.0-84.b.1.8 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0-3.a.1.2 | $84$ | $3$ | $3$ | $0$ |
| 84.24.0-6.a.1.2 | $84$ | $3$ | $3$ | $0$ |
| 84.32.1-12.a.1.4 | $84$ | $4$ | $4$ | $1$ |
| 84.64.1-21.a.1.9 | $84$ | $8$ | $8$ | $1$ |
| 84.168.5-21.a.1.14 | $84$ | $21$ | $21$ | $5$ |
| 84.224.6-21.a.1.15 | $84$ | $28$ | $28$ | $6$ |
| 168.16.0-24.a.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.16.0-168.a.1.9 | $168$ | $2$ | $2$ | $0$ |
| 168.16.0-24.b.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.16.0-168.b.1.11 | $168$ | $2$ | $2$ | $0$ |
| 168.16.0-24.c.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.16.0-168.c.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.16.0-24.d.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.16.0-168.d.1.15 | $168$ | $2$ | $2$ | $0$ |
| 252.24.0-9.a.1.4 | $252$ | $3$ | $3$ | $0$ |
| 252.24.0-9.b.1.4 | $252$ | $3$ | $3$ | $0$ |
| 252.24.1-9.a.1.4 | $252$ | $3$ | $3$ | $1$ |