Invariants
| Level: | $56$ | $\SL_2$-level: | $4$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $2^{2}$ | ||
| 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: | 4E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.24.0.44 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}7&36\\2&55\end{bmatrix}$, $\begin{bmatrix}18&39\\27&54\end{bmatrix}$, $\begin{bmatrix}23&28\\6&51\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.12.0.j.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $32$ |
| Cyclic 56-torsion field degree: | $768$ |
| Full 56-torsion field degree: | $129024$ |
Models
Smooth plane model Smooth plane model
| $ 0 $ | $=$ | $ 7 x^{2} + 7 y^{2} - 64 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 |
|---|---|---|---|---|---|
| 4.12.0-4.a.1.2 | $4$ | $2$ | $2$ | $0$ | $0$ |
| 56.12.0-4.a.1.1 | $56$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 56.192.5-56.s.1.10 | $56$ | $8$ | $8$ | $5$ |
| 56.504.16-56.bq.1.7 | $56$ | $21$ | $21$ | $16$ |
| 56.672.21-56.bq.1.3 | $56$ | $28$ | $28$ | $21$ |
| 168.72.2-168.bt.1.4 | $168$ | $3$ | $3$ | $2$ |
| 168.96.1-168.kf.1.13 | $168$ | $4$ | $4$ | $1$ |
| 280.120.4-280.v.1.7 | $280$ | $5$ | $5$ | $4$ |
| 280.144.3-280.bh.1.16 | $280$ | $6$ | $6$ | $3$ |
| 280.240.7-280.bt.1.10 | $280$ | $10$ | $10$ | $7$ |