Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | ||||
| 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: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.24.0.278 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}38&29\\33&0\end{bmatrix}$, $\begin{bmatrix}44&17\\19&34\end{bmatrix}$, $\begin{bmatrix}53&22\\18&41\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.12.0.h.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
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 40 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^5\,\frac{(x-4y)^{12}(7x^{4}-96x^{3}y+704x^{2}y^{2}-3072xy^{3}+7168y^{4})^{3}}{(x-4y)^{12}(x^{2}-16xy+32y^{2})^{4}(x^{2}-8xy+32y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 56.12.0-4.b.1.3 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.12.0-4.b.1.4 | $56$ | $2$ | $2$ | $0$ | $0$ |
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.n.1.12 | $56$ | $8$ | $8$ | $5$ |
| 56.504.16-56.z.1.4 | $56$ | $21$ | $21$ | $16$ |
| 56.672.21-56.z.1.5 | $56$ | $28$ | $28$ | $21$ |
| 112.48.0-16.c.1.2 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-16.c.1.4 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-112.c.1.3 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-112.c.1.8 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-16.d.1.1 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-16.d.1.2 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-112.d.1.3 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-112.d.1.8 | $112$ | $2$ | $2$ | $0$ |
| 168.72.2-24.z.1.5 | $168$ | $3$ | $3$ | $2$ |
| 168.96.1-24.dq.1.12 | $168$ | $4$ | $4$ | $1$ |
| 280.120.4-40.n.1.4 | $280$ | $5$ | $5$ | $4$ |
| 280.144.3-40.t.1.5 | $280$ | $6$ | $6$ | $3$ |
| 280.240.7-40.z.1.7 | $280$ | $10$ | $10$ | $7$ |