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: | $6$ (none of which are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | Cusp orbits | $2^{3}$ | ||
| 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: | 8G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.48.0.565 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}6&11\\9&28\end{bmatrix}$, $\begin{bmatrix}11&24\\18&13\end{bmatrix}$, $\begin{bmatrix}29&2\\20&27\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.24.0.v.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 26 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2}{7^4}\cdot\frac{(x-2y)^{24}(177x^{8}+4608x^{7}y+52192x^{6}y^{2}+258048x^{5}y^{3}+993664x^{4}y^{4}+2064384x^{3}y^{5}+3340288x^{2}y^{6}+2359296xy^{7}+724992y^{8})^{3}}{(x-2y)^{24}(x^{2}-8y^{2})^{8}(x^{2}+2xy+8y^{2})^{2}(x^{2}+16xy+8y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0-8.k.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0-8.k.1.3 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0-56.y.1.6 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0-56.y.1.7 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0-56.z.1.12 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0-56.z.1.14 | $56$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 56.384.11-56.cu.1.10 | $56$ | $8$ | $8$ | $11$ |
| 56.1008.34-56.dm.1.4 | $56$ | $21$ | $21$ | $34$ |
| 56.1344.45-56.do.1.9 | $56$ | $28$ | $28$ | $45$ |
| 168.144.4-168.fu.1.21 | $168$ | $3$ | $3$ | $4$ |
| 168.192.3-168.hi.1.19 | $168$ | $4$ | $4$ | $3$ |
| 280.240.8-280.cc.1.10 | $280$ | $5$ | $5$ | $8$ |
| 280.288.7-280.ee.1.32 | $280$ | $6$ | $6$ | $7$ |
| 280.480.15-280.fw.1.27 | $280$ | $10$ | $10$ | $15$ |