Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.48.0.616 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}4&41\\1&0\end{bmatrix}$, $\begin{bmatrix}23&28\\28&11\end{bmatrix}$, $\begin{bmatrix}26&53\\49&22\end{bmatrix}$, $\begin{bmatrix}33&26\\22&25\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.24.0.t.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $16$ |
| Cyclic 56-torsion field degree: | $192$ |
| Full 56-torsion field degree: | $64512$ |
Models
Smooth plane model Smooth plane model
| $ 0 $ | $=$ | $ 7 x^{2} - 2 y^{2} + 112 z^{2} $ |
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
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0-4.d.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0-4.d.1.1 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0-56.y.1.4 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0-56.y.1.5 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0-56.y.1.12 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0-56.y.1.13 | $56$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 56.96.1-56.dg.1.2 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.dh.1.2 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.di.1.2 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.dj.1.2 | $56$ | $2$ | $2$ | $1$ |
| 56.384.11-56.cs.1.2 | $56$ | $8$ | $8$ | $11$ |
| 56.1008.34-56.dk.1.3 | $56$ | $21$ | $21$ | $34$ |
| 56.1344.45-56.dm.1.5 | $56$ | $28$ | $28$ | $45$ |
| 168.96.1-168.ki.1.3 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.kj.1.3 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.kk.1.3 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.kl.1.3 | $168$ | $2$ | $2$ | $1$ |
| 168.144.4-168.fs.1.25 | $168$ | $3$ | $3$ | $4$ |
| 168.192.3-168.hg.1.21 | $168$ | $4$ | $4$ | $3$ |
| 280.96.1-280.jk.1.4 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.jl.1.4 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.jm.1.4 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.jn.1.4 | $280$ | $2$ | $2$ | $1$ |
| 280.240.8-280.ca.1.5 | $280$ | $5$ | $5$ | $8$ |
| 280.288.7-280.ec.1.5 | $280$ | $6$ | $6$ | $7$ |
| 280.480.15-280.fu.1.19 | $280$ | $10$ | $10$ | $15$ |