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: | $\begin{bmatrix}43&2\\52&51\end{bmatrix}$, $\begin{bmatrix}50&7\\33&10\end{bmatrix}$, $\begin{bmatrix}50&7\\41&36\end{bmatrix}$, $\begin{bmatrix}79&26\\40&33\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 84.24.0.n.1 for the level structure with $-I$) |
| Cyclic 84-isogeny field degree: | $16$ |
| Cyclic 84-torsion field degree: | $384$ |
| 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.1.8 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 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.a.1.14 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.e.1.10 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.q.1.6 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.s.1.6 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.bk.1.6 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.bm.1.7 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.bp.1.6 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.bq.1.2 | $84$ | $2$ | $2$ | $1$ |
| 84.144.1-84.m.1.3 | $84$ | $3$ | $3$ | $1$ |
| 84.384.11-84.cj.1.24 | $84$ | $8$ | $8$ | $11$ |
| 168.96.1-168.gf.1.14 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.jx.1.12 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bab.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bah.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.byu.1.14 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bza.1.12 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bzk.1.12 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bzn.1.12 | $168$ | $2$ | $2$ | $1$ |
| 252.144.1-252.g.1.6 | $252$ | $3$ | $3$ | $1$ |
| 252.144.4-252.q.1.13 | $252$ | $3$ | $3$ | $4$ |
| 252.144.4-252.bb.1.6 | $252$ | $3$ | $3$ | $4$ |