Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $6$ | ||||
| Genus: | $0 = 1 + \frac{ 6 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (of which $1$ is rational) | Cusp widths | $1^{2}\cdot4$ | Cusp orbits | $1\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 4B0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.12.0.22 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}1&52\\28&19\end{bmatrix}$, $\begin{bmatrix}9&20\\4&23\end{bmatrix}$, $\begin{bmatrix}20&27\\21&54\end{bmatrix}$, $\begin{bmatrix}24&3\\43&2\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 4.6.0.b.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $32$ |
| Cyclic 56-torsion field degree: | $768$ |
| Full 56-torsion field degree: | $258048$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 11629 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 6 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{6}(x^{2}+48y^{2})^{3}}{y^{4}x^{6}(x^{2}+64y^{2})}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 56.24.0-4.a.1.3 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-4.c.1.2 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-8.c.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-8.h.1.2 | $56$ | $2$ | $2$ | $0$ |
| 168.24.0-12.e.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-12.f.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.36.1-12.b.1.16 | $168$ | $3$ | $3$ | $1$ |
| 168.48.0-12.f.1.9 | $168$ | $4$ | $4$ | $0$ |
| 280.24.0-20.e.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-20.f.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.60.2-20.b.1.8 | $280$ | $5$ | $5$ | $2$ |
| 280.72.1-20.b.1.11 | $280$ | $6$ | $6$ | $1$ |
| 280.120.3-20.b.1.9 | $280$ | $10$ | $10$ | $3$ |
| 168.24.0-24.m.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-24.p.1.2 | $168$ | $2$ | $2$ | $0$ |
| 56.24.0-28.e.1.3 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-28.f.1.2 | $56$ | $2$ | $2$ | $0$ |
| 56.96.2-28.b.1.9 | $56$ | $8$ | $8$ | $2$ |
| 56.252.7-28.b.1.16 | $56$ | $21$ | $21$ | $7$ |
| 56.336.9-28.b.1.1 | $56$ | $28$ | $28$ | $9$ |
| 280.24.0-40.m.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-40.p.1.3 | $280$ | $2$ | $2$ | $0$ |
| 56.24.0-56.m.1.4 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-56.p.1.4 | $56$ | $2$ | $2$ | $0$ |
| 168.24.0-84.e.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-84.f.1.8 | $168$ | $2$ | $2$ | $0$ |
| 280.24.0-140.e.1.6 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-140.f.1.3 | $280$ | $2$ | $2$ | $0$ |
| 168.24.0-168.m.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.p.1.5 | $168$ | $2$ | $2$ | $0$ |
| 280.24.0-280.m.1.7 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-280.p.1.7 | $280$ | $2$ | $2$ | $0$ |