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 | $2^{3}\cdot6^{3}$ | 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: | 6I0 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}11&66\\38&73\end{bmatrix}$, $\begin{bmatrix}11&72\\42&53\end{bmatrix}$, $\begin{bmatrix}18&11\\77&36\end{bmatrix}$, $\begin{bmatrix}81&16\\28&45\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 42.24.0.c.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, including 53 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{12}\cdot3^3\cdot7}\cdot\frac{x^{24}(7x^{2}+144y^{2})^{3}(343x^{6}-35280x^{4}y^{2}+1209600x^{2}y^{4}+331776y^{6})^{3}}{y^{6}x^{26}(7x^{2}-432y^{2})^{2}(7x^{2}-48y^{2})^{6}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.24.0-6.a.1.6 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 84.24.0-6.a.1.6 | $84$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 84.144.1-42.e.1.4 | $84$ | $3$ | $3$ | $1$ |
| 84.384.11-42.l.1.16 | $84$ | $8$ | $8$ | $11$ |
| 84.96.1-84.m.1.1 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.o.1.4 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.bd.1.5 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.be.1.5 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.bl.1.2 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.bm.1.2 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.bs.1.6 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.bu.1.12 | $84$ | $2$ | $2$ | $1$ |
| 252.144.1-126.j.1.1 | $252$ | $3$ | $3$ | $1$ |
| 252.144.4-126.ba.1.1 | $252$ | $3$ | $3$ | $4$ |
| 252.144.4-126.be.1.11 | $252$ | $3$ | $3$ | $4$ |
| 168.96.1-168.zh.1.9 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zn.1.9 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.blk.1.3 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bln.1.3 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.byy.1.10 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bzb.1.10 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bzs.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bzy.1.4 | $168$ | $2$ | $2$ | $1$ |