Invariants
| Level: | $84$ | $\SL_2$-level: | $12$ | Newform level: | $144$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $12^{6}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12A4 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}9&76\\32&21\end{bmatrix}$, $\begin{bmatrix}29&54\\6&59\end{bmatrix}$, $\begin{bmatrix}61&22\\30&11\end{bmatrix}$, $\begin{bmatrix}83&72\\60&59\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.72.4.a.1 for the level structure with $-I$) |
| Cyclic 84-isogeny field degree: | $64$ |
| Cyclic 84-torsion field degree: | $1536$ |
| Full 84-torsion field degree: | $64512$ |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ x^{2} + y^{2} + z^{2} $ |
| $=$ | $x y z - 4 w^{3}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} y^{2} + x^{2} y^{4} + 16 z^{6} $ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=7$, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{y^{12}-48y^{6}w^{6}+y^{2}z^{10}-16y^{2}z^{4}w^{6}+z^{12}-32z^{6}w^{6}+768w^{12}}{w^{12}}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 12.72.4.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}Y^{2}+X^{2}Y^{4}+16Z^{6} $ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $48$ | $24$ | $0$ | $0$ |
| 28.48.0-4.a.1.1 | $28$ | $3$ | $3$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 28.48.0-4.a.1.1 | $28$ | $3$ | $3$ | $0$ | $0$ |
| 84.48.2-12.a.1.3 | $84$ | $3$ | $3$ | $2$ | $?$ |
| 84.72.2-12.a.1.1 | $84$ | $2$ | $2$ | $2$ | $?$ |
| 84.72.2-12.a.1.8 | $84$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.