Invariants
| Level: | $84$ | $\SL_2$-level: | $42$ | Newform level: | $294$ | ||
| Index: | $504$ | $\PSL_2$-index: | $252$ | ||||
| Genus: | $16 = 1 + \frac{ 252 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $7^{3}\cdot14^{3}\cdot21^{3}\cdot42^{3}$ | Cusp orbits | $3^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 30$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 16$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 42A16 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}1&44\\82&63\end{bmatrix}$, $\begin{bmatrix}3&58\\26&37\end{bmatrix}$, $\begin{bmatrix}46&77\\63&32\end{bmatrix}$, $\begin{bmatrix}70&57\\1&14\end{bmatrix}$, $\begin{bmatrix}70&75\\19&56\end{bmatrix}$, $\begin{bmatrix}73&8\\6&41\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 42.252.16.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: | $18432$ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(7)$ | $7$ | $24$ | $12$ | $0$ | $0$ |
| 12.24.0-6.a.1.9 | $12$ | $21$ | $21$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.24.0-6.a.1.9 | $12$ | $21$ | $21$ | $0$ | $0$ |
| 84.168.5-21.a.1.5 | $84$ | $3$ | $3$ | $5$ | $?$ |