Invariants
| Level: | $84$ | $\SL_2$-level: | $4$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $6$ | ||||
| Genus: | $0 = 1 + \frac{ 6 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (all of which are rational) | Cusp widths | $1^{2}\cdot4$ | Cusp orbits | $1^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4B0 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}8&3\\67&80\end{bmatrix}$, $\begin{bmatrix}10&29\\39&28\end{bmatrix}$, $\begin{bmatrix}77&6\\60&43\end{bmatrix}$, $\begin{bmatrix}77&80\\54&79\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 4.6.0.c.1 for the level structure with $-I$) |
| Cyclic 84-isogeny field degree: | $32$ |
| Cyclic 84-torsion field degree: | $768$ |
| Full 84-torsion field degree: | $774144$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 95098 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 6 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{6}(48x^{2}-y^{2})^{3}}{x^{10}(8x-y)(8x+y)}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 84.24.0-4.b.1.3 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0-4.d.1.1 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0-12.g.1.2 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0-28.g.1.2 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0-84.g.1.2 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0-12.h.1.1 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0-28.h.1.1 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0-84.h.1.1 | $84$ | $2$ | $2$ | $0$ |
| 84.36.1-12.c.1.6 | $84$ | $3$ | $3$ | $1$ |
| 84.48.0-12.g.1.4 | $84$ | $4$ | $4$ | $0$ |
| 84.96.2-28.c.1.1 | $84$ | $8$ | $8$ | $2$ |
| 84.252.7-28.c.1.4 | $84$ | $21$ | $21$ | $7$ |
| 84.336.9-28.c.1.1 | $84$ | $28$ | $28$ | $9$ |
| 168.24.0-8.d.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-8.k.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-8.m.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-8.m.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-8.n.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-8.n.1.12 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-8.o.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-8.o.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-8.p.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-8.p.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-24.s.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-56.s.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.s.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-24.v.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-56.v.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.v.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-24.y.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-24.y.1.15 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-56.y.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-56.y.1.16 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.y.1.16 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.y.1.17 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-24.z.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-24.z.1.11 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-56.z.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-56.z.1.14 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.z.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.z.1.25 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-24.ba.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-24.ba.1.11 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-56.ba.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-56.ba.1.14 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.ba.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.ba.1.25 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-24.bb.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-24.bb.1.15 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-56.bb.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-56.bb.1.16 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.bb.1.16 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.bb.1.17 | $168$ | $2$ | $2$ | $0$ |
| 252.324.10-36.d.1.1 | $252$ | $27$ | $27$ | $10$ |