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: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 4E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.24.0.34 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}11&40\\18&11\end{bmatrix}$, $\begin{bmatrix}17&46\\10&7\end{bmatrix}$, $\begin{bmatrix}25&42\\16&3\end{bmatrix}$, $\begin{bmatrix}29&44\\32&41\end{bmatrix}$, $\begin{bmatrix}35&40\\34&31\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 4.12.0.a.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 746 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{12}(x^{2}-4xy+16y^{2})^{3}(x^{2}+4xy+16y^{2})^{3}}{y^{4}x^{16}(x^{2}+16y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 56.12.0-4.b.1.1 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.12.0-4.b.1.3 | $56$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 56.48.0-4.a.1.3 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-8.a.1.2 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-28.a.1.6 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.a.1.5 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-4.b.1.7 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-8.b.1.2 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-28.b.1.6 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.c.1.5 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-8.f.1.2 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-8.g.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.j.1.3 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.k.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.48.1-8.a.1.4 | $56$ | $2$ | $2$ | $1$ |
| 56.48.1-56.a.1.7 | $56$ | $2$ | $2$ | $1$ |
| 56.48.1-8.b.1.2 | $56$ | $2$ | $2$ | $1$ |
| 56.48.1-56.b.1.2 | $56$ | $2$ | $2$ | $1$ |
| 56.192.5-28.a.1.9 | $56$ | $8$ | $8$ | $5$ |
| 56.504.16-28.a.1.18 | $56$ | $21$ | $21$ | $16$ |
| 56.672.21-28.a.1.15 | $56$ | $28$ | $28$ | $21$ |
| 168.48.0-12.a.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.a.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-84.a.1.12 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.a.1.11 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-12.b.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-84.b.1.12 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.c.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.c.1.11 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.j.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.k.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.v.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.w.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.1-24.a.1.6 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-168.a.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-24.b.1.6 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-168.b.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.72.2-12.a.1.16 | $168$ | $3$ | $3$ | $2$ |
| 168.96.1-12.a.1.18 | $168$ | $4$ | $4$ | $1$ |
| 280.48.0-20.a.1.4 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.a.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-140.a.1.9 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.a.1.7 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-20.b.1.8 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-140.b.1.10 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.c.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.c.1.7 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.j.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.k.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.v.1.7 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.w.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.48.1-40.a.1.2 | $280$ | $2$ | $2$ | $1$ |
| 280.48.1-280.a.1.16 | $280$ | $2$ | $2$ | $1$ |
| 280.48.1-40.b.1.4 | $280$ | $2$ | $2$ | $1$ |
| 280.48.1-280.b.1.6 | $280$ | $2$ | $2$ | $1$ |
| 280.120.4-20.a.1.10 | $280$ | $5$ | $5$ | $4$ |
| 280.144.3-20.a.1.11 | $280$ | $6$ | $6$ | $3$ |
| 280.240.7-20.a.1.22 | $280$ | $10$ | $10$ | $7$ |