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: | $6$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{3}\cdot8$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8J0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.48.0.239 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}1&16\\40&43\end{bmatrix}$, $\begin{bmatrix}39&8\\20&43\end{bmatrix}$, $\begin{bmatrix}39&24\\2&13\end{bmatrix}$, $\begin{bmatrix}41&4\\44&23\end{bmatrix}$, $\begin{bmatrix}45&32\\52&49\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.24.0.i.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $16$ |
| Cyclic 56-torsion field degree: | $384$ |
| Full 56-torsion field degree: | $64512$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 63 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^8}{3^8\cdot7^2}\cdot\frac{x^{24}(2401x^{8}+12348x^{6}y^{2}+19845x^{4}y^{4}+10206x^{2}y^{6}+6561y^{8})^{3}}{y^{8}x^{28}(7x^{2}+9y^{2})^{4}(7x^{2}+18y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0-4.b.1.10 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0-4.b.1.5 | $56$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.