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 | $2^{4}\cdot8^{2}$ | 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: | 8G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.48.0.774 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}2&13\\25&22\end{bmatrix}$, $\begin{bmatrix}34&35\\41&24\end{bmatrix}$, $\begin{bmatrix}36&5\\41&32\end{bmatrix}$, $\begin{bmatrix}52&33\\43&30\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.24.0.bl.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $8$ |
| Cyclic 56-torsion field degree: | $192$ |
| 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 25 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2}{3^2\cdot7}\cdot\frac{x^{24}(2401x^{8}-370440x^{6}y^{2}+2127384x^{4}y^{4}-2449440x^{2}y^{6}+104976y^{8})^{3}}{y^{2}x^{26}(7x^{2}-18y^{2})^{2}(7x^{2}+18y^{2})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0-8.n.1.11 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0-8.n.1.10 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0-56.v.1.3 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0-56.v.1.5 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0-56.y.1.1 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0-56.y.1.3 | $56$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.