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: | $\begin{bmatrix}9&8\\11&29\end{bmatrix}$, $\begin{bmatrix}47&72\\51&67\end{bmatrix}$, $\begin{bmatrix}55&20\\5&21\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.12.0.g.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 621 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{3}\cdot\frac{(12x+y)^{12}(48x^{2}-24xy-y^{2})^{3}(48x^{2}+24xy-y^{2})^{3}}{y^{2}x^{2}(12x+y)^{12}(48x^{2}+y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 28.12.0-4.c.1.2 | $28$ | $2$ | $2$ | $0$ | $0$ |
| 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 |
|---|---|---|---|---|
| 84.72.2-12.s.1.1 | $84$ | $3$ | $3$ | $2$ |
| 84.96.1-12.k.1.1 | $84$ | $4$ | $4$ | $1$ |
| 84.192.5-84.k.1.2 | $84$ | $8$ | $8$ | $5$ |
| 84.504.16-84.s.1.2 | $84$ | $21$ | $21$ | $16$ |
| 168.48.0-24.bg.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bg.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bh.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bh.1.9 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bo.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bo.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bp.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bp.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.bq.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.bq.1.14 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.br.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.br.1.11 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.bu.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.bu.1.10 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.bv.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.bv.1.12 | $168$ | $2$ | $2$ | $0$ |