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: | $\begin{bmatrix}22&29\\45&62\end{bmatrix}$, $\begin{bmatrix}31&36\\82&17\end{bmatrix}$, $\begin{bmatrix}40&27\\75&28\end{bmatrix}$, $\begin{bmatrix}70&47\\69&38\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 42.24.0.b.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 96 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{5^3}{7}\cdot\frac{(5x+3y)^{24}(20x^{2}-6xy-9y^{2})^{3}(25000000x^{6}+78300000x^{5}y+105300000x^{4}y^{2}+76275000x^{3}y^{3}+30861000x^{2}y^{4}+6466230xy^{5}+538731y^{6})^{3}}{(5x+3y)^{24}(10x+3y)^{2}(10x+9y)^{6}(100x^{2}+75xy+18y^{2})^{6}(200x^{2}+255xy+99y^{2})^{2}}$ |
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.1.10 | $84$ | $2$ | $2$ | $0$ | $?$ |
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.d.1.6 | $84$ | $3$ | $3$ | $1$ |
| 84.384.11-42.k.1.11 | $84$ | $8$ | $8$ | $11$ |
| 84.96.1-84.i.1.1 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.k.1.9 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.u.1.7 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.w.1.5 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.bg.1.5 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.bi.1.2 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.bo.1.5 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.bq.1.12 | $84$ | $2$ | $2$ | $1$ |
| 252.144.1-126.i.1.1 | $252$ | $3$ | $3$ | $1$ |
| 252.144.4-126.w.1.1 | $252$ | $3$ | $3$ | $4$ |
| 252.144.4-126.bd.1.2 | $252$ | $3$ | $3$ | $4$ |
| 168.96.1-168.yv.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zb.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.ban.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bat.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.byi.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.byo.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bzg.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bzm.1.4 | $168$ | $2$ | $2$ | $1$ |