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: | $\begin{bmatrix}33&16\\71&23\end{bmatrix}$, $\begin{bmatrix}35&44\\61&29\end{bmatrix}$, $\begin{bmatrix}81&52\\29&37\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 28.12.0.h.1 for the level structure with $-I$) |
| Cyclic 84-isogeny field degree: | $32$ |
| Cyclic 84-torsion field degree: | $768$ |
| 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, including 556 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^2\cdot7}\cdot\frac{(28x-y)^{12}(200704x^{4}+6272x^{2}y^{2}+y^{4})^{3}}{y^{2}x^{2}(28x-y)^{12}(448x^{2}-y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.12.0-4.c.1.1 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 84.12.0-4.c.1.2 | $84$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 84.72.2-84.t.1.7 | $84$ | $3$ | $3$ | $2$ |
| 84.96.1-84.l.1.8 | $84$ | $4$ | $4$ | $1$ |
| 84.192.5-28.l.1.2 | $84$ | $8$ | $8$ | $5$ |
| 84.504.16-28.t.1.2 | $84$ | $21$ | $21$ | $16$ |
| 168.48.0-56.bi.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.bi.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.bj.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.bj.1.11 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.bq.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.bq.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.br.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.br.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.ck.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.ck.1.16 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.cl.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.cl.1.15 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.cs.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.cs.1.15 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.ct.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.ct.1.16 | $168$ | $2$ | $2$ | $0$ |