Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $2^{3}\cdot4$ | ||
| 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: | 8O0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.96.0.343 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}23&24\\24&27\end{bmatrix}$, $\begin{bmatrix}39&34\\38&7\end{bmatrix}$, $\begin{bmatrix}45&12\\34&19\end{bmatrix}$, $\begin{bmatrix}51&50\\12&45\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.0.x.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $16$ |
| Cyclic 56-torsion field degree: | $192$ |
| Full 56-torsion field degree: | $32256$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 5 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{12}\cdot5^8\cdot7^2}\cdot\frac{(4x-3y)^{48}(692224x^{8}-3067904x^{7}y+13447168x^{6}y^{2}-38610432x^{5}y^{3}+57796480x^{4}y^{4}-44540608x^{3}y^{5}+31808448x^{2}y^{6}-25296936xy^{7}+11354329y^{8})^{3}(19369984x^{8}-37761024x^{7}y+41545728x^{6}y^{2}-50903552x^{5}y^{3}+57796480x^{4}y^{4}-33784128x^{3}y^{5}+10295488x^{2}y^{6}-2055256xy^{7}+405769y^{8})^{3}}{(4x-3y)^{48}(8x^{2}-7y^{2})^{8}(8x^{2}-32xy+7y^{2})^{4}(8x^{2}-7xy+7y^{2})^{8}(3648x^{4}-3584x^{3}y+2352x^{2}y^{2}-3136xy^{3}+2793y^{4})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.h.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-8.h.1.7 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.h.1.17 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.h.1.31 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.i.1.23 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.i.1.25 | $56$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.