Invariants
| Level: | $84$ | $\SL_2$-level: | $12$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot3^{2}\cdot4\cdot12$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| 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: | 12E0 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}26&3\\27&56\end{bmatrix}$, $\begin{bmatrix}43&74\\0&11\end{bmatrix}$, $\begin{bmatrix}52&71\\1&48\end{bmatrix}$, $\begin{bmatrix}61&12\\28&65\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 84.24.0.r.1 for the level structure with $-I$) |
| Cyclic 84-isogeny field degree: | $16$ |
| Cyclic 84-torsion field degree: | $192$ |
| Full 84-torsion field degree: | $193536$ |
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 |
|---|---|---|---|---|---|
| 12.24.0-6.a.1.11 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 42.24.0-6.a.1.4 | $42$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 84.96.1-84.d.1.19 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.g.1.9 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.y.1.7 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.bb.1.5 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.bg.1.1 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.bj.1.5 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.bs.1.7 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.bv.1.9 | $84$ | $2$ | $2$ | $1$ |
| 84.144.1-84.q.1.2 | $84$ | $3$ | $3$ | $1$ |
| 84.384.11-84.cn.1.24 | $84$ | $8$ | $8$ | $11$ |
| 168.96.1-168.gi.1.14 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.kd.1.12 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bkv.1.14 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.ble.1.14 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.byj.1.14 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bys.1.14 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bzt.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.cac.1.12 | $168$ | $2$ | $2$ | $1$ |
| 252.144.1-252.k.1.14 | $252$ | $3$ | $3$ | $1$ |
| 252.144.4-252.x.1.5 | $252$ | $3$ | $3$ | $4$ |
| 252.144.4-252.bf.1.6 | $252$ | $3$ | $3$ | $4$ |