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: | $\begin{bmatrix}13&60\\18&13\end{bmatrix}$, $\begin{bmatrix}36&19\\37&12\end{bmatrix}$, $\begin{bmatrix}50&7\\27&22\end{bmatrix}$, $\begin{bmatrix}62&31\\41&48\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 84.24.0.q.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.1.9 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 84.24.0-6.a.1.11 | $84$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 84.96.1-84.c.1.7 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.f.1.2 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.z.1.6 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.ba.1.7 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.bh.1.6 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.bi.1.3 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.bt.1.5 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.bu.1.10 | $84$ | $2$ | $2$ | $1$ |
| 84.144.1-84.r.1.3 | $84$ | $3$ | $3$ | $1$ |
| 84.384.11-84.cm.1.16 | $84$ | $8$ | $8$ | $11$ |
| 168.96.1-168.gh.1.15 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.ka.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bky.1.15 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.blb.1.14 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bym.1.6 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.byp.1.15 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bzw.1.6 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bzz.1.15 | $168$ | $2$ | $2$ | $1$ |
| 252.144.1-252.j.1.10 | $252$ | $3$ | $3$ | $1$ |
| 252.144.4-252.w.1.7 | $252$ | $3$ | $3$ | $4$ |
| 252.144.4-252.be.1.2 | $252$ | $3$ | $3$ | $4$ |