Invariants
| Level: | $84$ | $\SL_2$-level: | $28$ | Newform level: | $1$ | ||
| Index: | $168$ | $\PSL_2$-index: | $168$ | ||||
| Genus: | $9 = 1 + \frac{ 168 }{12} - \frac{ 8 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $2$ are rational) | Cusp widths | $14^{4}\cdot28^{4}$ | Cusp orbits | $1^{2}\cdot3^{2}$ | ||
| Elliptic points: | $8$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 9$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 9$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 28B9 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}6&11\\73&66\end{bmatrix}$, $\begin{bmatrix}26&57\\55&76\end{bmatrix}$, $\begin{bmatrix}32&11\\43&66\end{bmatrix}$, $\begin{bmatrix}34&21\\83&22\end{bmatrix}$, $\begin{bmatrix}70&27\\29&34\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 84-isogeny field degree: | $16$ |
| Cyclic 84-torsion field degree: | $384$ |
| Full 84-torsion field degree: | $55296$ |
Rational points
This modular curve has 2 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 |
|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(7)$ | $7$ | $6$ | $6$ | $0$ | $0$ |
| 12.6.0.f.1 | $12$ | $28$ | $28$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.6.0.f.1 | $12$ | $28$ | $28$ | $0$ | $0$ |
| 14.84.3.a.1 | $14$ | $2$ | $2$ | $3$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 84.336.17.fd.1 | $84$ | $2$ | $2$ | $17$ |
| 84.336.17.fe.1 | $84$ | $2$ | $2$ | $17$ |
| 84.336.17.fh.1 | $84$ | $2$ | $2$ | $17$ |
| 84.336.17.fi.1 | $84$ | $2$ | $2$ | $17$ |
| 84.336.17.fq.1 | $84$ | $2$ | $2$ | $17$ |
| 84.336.17.fr.1 | $84$ | $2$ | $2$ | $17$ |
| 84.336.17.fu.1 | $84$ | $2$ | $2$ | $17$ |
| 84.336.17.fv.1 | $84$ | $2$ | $2$ | $17$ |
| 84.336.21.b.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.i.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.r.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.s.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.fe.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.ff.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.fi.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.fj.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.ha.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.hb.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.he.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.hf.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.ik.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.il.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.io.1 | $84$ | $2$ | $2$ | $21$ |
| 84.336.21.ip.1 | $84$ | $2$ | $2$ | $21$ |
| 168.336.17.rg.1 | $168$ | $2$ | $2$ | $17$ |
| 168.336.17.rj.1 | $168$ | $2$ | $2$ | $17$ |
| 168.336.17.rs.1 | $168$ | $2$ | $2$ | $17$ |
| 168.336.17.rv.1 | $168$ | $2$ | $2$ | $17$ |
| 168.336.17.ss.1 | $168$ | $2$ | $2$ | $17$ |
| 168.336.17.sv.1 | $168$ | $2$ | $2$ | $17$ |
| 168.336.17.te.1 | $168$ | $2$ | $2$ | $17$ |
| 168.336.17.th.1 | $168$ | $2$ | $2$ | $17$ |
| 168.336.21.w.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.ba.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.cb.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.ce.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.qc.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.qf.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.qo.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.qr.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.wc.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.wf.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.wo.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.wr.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.bbe.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.bbh.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.bbq.1 | $168$ | $2$ | $2$ | $21$ |
| 168.336.21.bbt.1 | $168$ | $2$ | $2$ | $21$ |