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: | $6$ (all of which are rational) | Cusp widths | $2^{3}\cdot6^{3}$ | Cusp orbits | $1^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $6$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6I0 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}11&36\\66&65\end{bmatrix}$, $\begin{bmatrix}51&62\\64&29\end{bmatrix}$, $\begin{bmatrix}55&78\\78&73\end{bmatrix}$, $\begin{bmatrix}57&38\\8&63\end{bmatrix}$, $\begin{bmatrix}77&30\\12&47\end{bmatrix}$, $\begin{bmatrix}81&4\\20&61\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 6.24.0.a.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 110 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6}\cdot\frac{x^{24}(x^{2}+12y^{2})^{3}(x^{6}-60x^{4}y^{2}+1200x^{2}y^{4}+192y^{6})^{3}}{y^{6}x^{26}(x-6y)^{2}(x-2y)^{6}(x+2y)^{6}(x+6y)^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 84.24.0-6.a.1.7 | $84$ | $2$ | $2$ | $0$ | $?$ |
| 84.24.0-6.a.1.9 | $84$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 84.96.0-12.a.1.5 | $84$ | $2$ | $2$ | $0$ |
| 84.96.0-12.a.1.10 | $84$ | $2$ | $2$ | $0$ |
| 84.96.0-12.a.1.16 | $84$ | $2$ | $2$ | $0$ |
| 84.96.0-12.a.2.2 | $84$ | $2$ | $2$ | $0$ |
| 84.96.0-12.a.2.11 | $84$ | $2$ | $2$ | $0$ |
| 84.96.0-12.a.2.16 | $84$ | $2$ | $2$ | $0$ |
| 84.96.0-84.a.1.12 | $84$ | $2$ | $2$ | $0$ |
| 84.96.0-84.a.1.15 | $84$ | $2$ | $2$ | $0$ |
| 84.96.0-84.a.1.21 | $84$ | $2$ | $2$ | $0$ |
| 84.96.0-84.a.2.1 | $84$ | $2$ | $2$ | $0$ |
| 84.96.0-84.a.2.23 | $84$ | $2$ | $2$ | $0$ |
| 84.96.0-84.a.2.28 | $84$ | $2$ | $2$ | $0$ |
| 84.96.1-12.a.1.6 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-12.a.1.10 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.a.1.8 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.a.1.11 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-12.b.1.5 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-12.b.1.12 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.b.1.8 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.b.1.17 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-12.c.1.3 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-12.c.1.9 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.c.1.5 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.c.1.11 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-12.d.1.7 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-12.d.1.8 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.d.1.1 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.d.1.20 | $84$ | $2$ | $2$ | $1$ |
| 84.96.2-12.a.1.5 | $84$ | $2$ | $2$ | $2$ |
| 84.96.2-12.a.2.5 | $84$ | $2$ | $2$ | $2$ |
| 84.96.2-84.a.1.9 | $84$ | $2$ | $2$ | $2$ |
| 84.96.2-84.a.2.2 | $84$ | $2$ | $2$ | $2$ |
| 84.144.1-6.a.1.4 | $84$ | $3$ | $3$ | $1$ |
| 84.384.11-42.a.1.15 | $84$ | $8$ | $8$ | $11$ |
| 168.96.0-24.o.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.o.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.o.1.26 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.o.2.1 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.o.2.12 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.o.2.18 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.o.1.48 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.o.1.50 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.o.1.57 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.o.2.48 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.o.2.50 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.o.2.57 | $168$ | $2$ | $2$ | $0$ |
| 168.96.1-24.bw.1.6 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.bw.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.bx.1.6 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.bx.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.by.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.by.1.6 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.bz.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.bz.1.6 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.dg.1.34 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.dg.1.35 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.dh.1.31 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.dh.1.38 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.di.1.31 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.di.1.38 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.dj.1.34 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.dj.1.35 | $168$ | $2$ | $2$ | $1$ |
| 168.96.2-24.b.1.3 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-24.b.2.5 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.b.1.24 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.b.2.24 | $168$ | $2$ | $2$ | $2$ |
| 252.144.1-18.a.1.7 | $252$ | $3$ | $3$ | $1$ |
| 252.144.4-18.a.1.7 | $252$ | $3$ | $3$ | $4$ |
| 252.144.4-18.b.1.8 | $252$ | $3$ | $3$ | $4$ |