Invariants
| Level: | $56$ | $\SL_2$-level: | $4$ | ||||
| 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 | $4^{6}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.48.0.102 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}1&8\\30&39\end{bmatrix}$, $\begin{bmatrix}27&30\\4&1\end{bmatrix}$, $\begin{bmatrix}55&28\\22&27\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.24.0.g.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $32$ |
| Cyclic 56-torsion field degree: | $384$ |
| Full 56-torsion field degree: | $64512$ |
Models
Smooth plane model Smooth plane model
| $ 0 $ | $=$ | $ 2 x^{2} - 3 x y + 2 y^{2} + 56 z^{2} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0-8.b.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0-8.b.1.4 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 28.24.0-28.b.1.3 | $28$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0-28.b.1.1 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0-56.a.1.4 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0-56.a.1.8 | $56$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 56.384.11-56.j.1.4 | $56$ | $8$ | $8$ | $11$ |
| 56.1008.34-56.j.1.10 | $56$ | $21$ | $21$ | $34$ |
| 56.1344.45-56.j.1.2 | $56$ | $28$ | $28$ | $45$ |
| 168.144.4-168.g.1.31 | $168$ | $3$ | $3$ | $4$ |
| 168.192.3-168.cw.1.25 | $168$ | $4$ | $4$ | $3$ |
| 280.240.8-280.g.1.16 | $280$ | $5$ | $5$ | $8$ |
| 280.288.7-280.i.1.29 | $280$ | $6$ | $6$ | $7$ |
| 280.480.15-280.g.1.29 | $280$ | $10$ | $10$ | $15$ |