Invariants
| Level: | $84$ | $\SL_2$-level: | $28$ | Newform level: | $1$ | ||
| Index: | $336$ | $\PSL_2$-index: | $336$ | ||||
| Genus: | $21 = 1 + \frac{ 336 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $4$ are rational) | Cusp widths | $14^{8}\cdot28^{8}$ | Cusp orbits | $1^{4}\cdot6^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $5 \le \gamma \le 21$ | ||||||
| $\overline{\Q}$-gonality: | $5 \le \gamma \le 21$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 28E21 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}12&5\\23&10\end{bmatrix}$, $\begin{bmatrix}61&40\\2&3\end{bmatrix}$, $\begin{bmatrix}67&62\\36&59\end{bmatrix}$, $\begin{bmatrix}70&51\\5&44\end{bmatrix}$, $\begin{bmatrix}80&3\\67&62\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 84-isogeny field degree: | $8$ |
| Cyclic 84-torsion field degree: | $192$ |
| Full 84-torsion field degree: | $27648$ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal 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 |
|---|---|---|---|---|---|
| $X_{\mathrm{sp}}(7)$ | $7$ | $6$ | $6$ | $1$ | $0$ |
| 12.6.0.f.1 | $12$ | $56$ | $56$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 14.168.7.a.1 | $14$ | $2$ | $2$ | $7$ | $0$ |
| 84.48.3.ce.1 | $84$ | $7$ | $7$ | $3$ | $?$ |
| 84.168.9.bg.1 | $84$ | $2$ | $2$ | $9$ | $?$ |
| 84.168.11.b.1 | $84$ | $2$ | $2$ | $11$ | $?$ |