Invariants
| Level: | $56$ | $\SL_2$-level: | $2$ | ||||
| Index: | $6$ | $\PSL_2$-index: | $6$ | ||||
| Genus: | $0 = 1 + \frac{ 6 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (of which $1$ is rational) | Cusp widths | $2^{3}$ | Cusp orbits | $1\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 2C0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.6.0.6 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}12&15\\41&40\end{bmatrix}$, $\begin{bmatrix}15&44\\24&19\end{bmatrix}$, $\begin{bmatrix}32&7\\19&6\end{bmatrix}$, $\begin{bmatrix}51&20\\22&55\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 56-isogeny field degree: | $32$ |
| Cyclic 56-torsion field degree: | $768$ |
| Full 56-torsion field degree: | $516096$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 2940 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 6 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^6}{3^2\cdot7}\cdot\frac{(48x-y)^{6}(52992x^{2}+5856xy+191y^{2})^{3}}{(16x+y)^{2}(48x-y)^{6}(11520x^{2}+2208xy+61y^{2})^{2}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(2)$ | $2$ | $2$ | $2$ | $0$ | $0$ |
| 56.2.0.a.1 | $56$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 56.12.0.m.1 | $56$ | $2$ | $2$ | $0$ |
| 56.12.0.o.1 | $56$ | $2$ | $2$ | $0$ |
| 56.12.0.s.1 | $56$ | $2$ | $2$ | $0$ |
| 56.12.0.u.1 | $56$ | $2$ | $2$ | $0$ |
| 56.12.0.bd.1 | $56$ | $2$ | $2$ | $0$ |
| 56.12.0.be.1 | $56$ | $2$ | $2$ | $0$ |
| 56.12.0.bj.1 | $56$ | $2$ | $2$ | $0$ |
| 56.12.0.bk.1 | $56$ | $2$ | $2$ | $0$ |
| 56.48.2.l.1 | $56$ | $8$ | $8$ | $2$ |
| 56.126.7.e.1 | $56$ | $21$ | $21$ | $7$ |
| 56.168.9.q.1 | $56$ | $28$ | $28$ | $9$ |
| 168.12.0.bo.1 | $168$ | $2$ | $2$ | $0$ |
| 168.12.0.bq.1 | $168$ | $2$ | $2$ | $0$ |
| 168.12.0.bu.1 | $168$ | $2$ | $2$ | $0$ |
| 168.12.0.bw.1 | $168$ | $2$ | $2$ | $0$ |
| 168.12.0.db.1 | $168$ | $2$ | $2$ | $0$ |
| 168.12.0.dc.1 | $168$ | $2$ | $2$ | $0$ |
| 168.12.0.dh.1 | $168$ | $2$ | $2$ | $0$ |
| 168.12.0.di.1 | $168$ | $2$ | $2$ | $0$ |
| 168.18.1.a.1 | $168$ | $3$ | $3$ | $1$ |
| 168.24.0.fg.1 | $168$ | $4$ | $4$ | $0$ |
| 280.12.0.bo.1 | $280$ | $2$ | $2$ | $0$ |
| 280.12.0.bq.1 | $280$ | $2$ | $2$ | $0$ |
| 280.12.0.bu.1 | $280$ | $2$ | $2$ | $0$ |
| 280.12.0.bw.1 | $280$ | $2$ | $2$ | $0$ |
| 280.12.0.db.1 | $280$ | $2$ | $2$ | $0$ |
| 280.12.0.dc.1 | $280$ | $2$ | $2$ | $0$ |
| 280.12.0.dh.1 | $280$ | $2$ | $2$ | $0$ |
| 280.12.0.di.1 | $280$ | $2$ | $2$ | $0$ |
| 280.30.2.a.1 | $280$ | $5$ | $5$ | $2$ |
| 280.36.1.a.1 | $280$ | $6$ | $6$ | $1$ |
| 280.60.3.ds.1 | $280$ | $10$ | $10$ | $3$ |