Invariants
| Level: | $84$ | $\SL_2$-level: | $28$ | Newform level: | $784$ | ||
| 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 | $14^{6}\cdot28^{6}$ | Cusp orbits | $3^{2}\cdot6$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $5 \le \gamma \le 30$ | ||||||
| $\overline{\Q}$-gonality: | $5 \le \gamma \le 16$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 28C16 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}13&51\\76&71\end{bmatrix}$, $\begin{bmatrix}23&39\\56&61\end{bmatrix}$, $\begin{bmatrix}31&23\\4&25\end{bmatrix}$, $\begin{bmatrix}59&60\\64&19\end{bmatrix}$, $\begin{bmatrix}83&44\\4&35\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 28.252.16.p.1 for the level structure with $-I$) |
| Cyclic 84-isogeny field degree: | $32$ |
| Cyclic 84-torsion field degree: | $768$ |
| 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 |
|---|---|---|---|---|---|
| 12.24.0-4.d.1.2 | $12$ | $21$ | $21$ | $0$ | $0$ |
| $X_{\mathrm{ns}}^+(7)$ | $7$ | $24$ | $12$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.24.0-4.d.1.2 | $12$ | $21$ | $21$ | $0$ | $0$ |
| 84.252.7-28.a.1.1 | $84$ | $2$ | $2$ | $7$ | $?$ |
| 84.252.7-28.a.1.8 | $84$ | $2$ | $2$ | $7$ | $?$ |