Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $112$ | ||
| Index: | $768$ | $\PSL_2$-index: | $768$ | ||||
| Genus: | $45 = 1 + \frac{ 768 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 40 }{2}$ | ||||||
| Cusps: | $40$ (of which $8$ are rational) | Cusp widths | $4^{16}\cdot8^{4}\cdot28^{16}\cdot56^{4}$ | Cusp orbits | $1^{8}\cdot2^{6}\cdot4^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $9 \le \gamma \le 16$ | ||||||
| $\overline{\Q}$-gonality: | $9 \le \gamma \le 16$ | ||||||
| Rational cusps: | $8$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.768.45.34 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}11&48\\40&47\end{bmatrix}$, $\begin{bmatrix}27&8\\52&9\end{bmatrix}$, $\begin{bmatrix}27&16\\36&41\end{bmatrix}$, $\begin{bmatrix}31&4\\28&9\end{bmatrix}$, $\begin{bmatrix}31&48\\12&39\end{bmatrix}$, $\begin{bmatrix}43&36\\8&31\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 56.1536.45-56.g.1.1, 56.1536.45-56.g.1.2, 56.1536.45-56.g.1.3, 56.1536.45-56.g.1.4, 56.1536.45-56.g.1.5, 56.1536.45-56.g.1.6, 56.1536.45-56.g.1.7, 56.1536.45-56.g.1.8, 56.1536.45-56.g.1.9, 56.1536.45-56.g.1.10, 56.1536.45-56.g.1.11, 56.1536.45-56.g.1.12, 56.1536.45-56.g.1.13, 56.1536.45-56.g.1.14, 56.1536.45-56.g.1.15, 56.1536.45-56.g.1.16, 56.1536.45-56.g.1.17, 56.1536.45-56.g.1.18, 56.1536.45-56.g.1.19, 56.1536.45-56.g.1.20 |
| Cyclic 56-isogeny field degree: | $2$ |
| Cyclic 56-torsion field degree: | $12$ |
| Full 56-torsion field degree: | $4032$ |
Jacobian
| Conductor: | $2^{128}\cdot7^{45}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{11}\cdot2^{9}\cdot4^{4}$ |
| Newforms: | 14.2.a.a$^{4}$, 28.2.d.a$^{3}$, 56.2.a.a$^{2}$, 56.2.a.b$^{2}$, 56.2.b.a$^{2}$, 56.2.b.b$^{2}$, 56.2.e.a$^{2}$, 56.2.e.b$^{2}$, 112.2.a.a, 112.2.a.b, 112.2.a.c, 112.2.f.a, 112.2.f.b |
Rational points
This modular curve has 8 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 28.384.21.b.1 | $28$ | $2$ | $2$ | $21$ | $1$ | $2^{4}\cdot4^{4}$ |
| 56.384.21.bc.3 | $56$ | $2$ | $2$ | $21$ | $0$ | $1^{6}\cdot2^{5}\cdot4^{2}$ |
| 56.384.21.bc.4 | $56$ | $2$ | $2$ | $21$ | $0$ | $1^{6}\cdot2^{5}\cdot4^{2}$ |
| 56.384.23.a.2 | $56$ | $2$ | $2$ | $23$ | $1$ | $2^{7}\cdot4^{2}$ |
| 56.384.23.i.1 | $56$ | $2$ | $2$ | $23$ | $1$ | $2^{7}\cdot4^{2}$ |
| 56.384.23.cd.2 | $56$ | $2$ | $2$ | $23$ | $0$ | $1^{6}\cdot2^{4}\cdot4^{2}$ |
| 56.384.23.cd.4 | $56$ | $2$ | $2$ | $23$ | $0$ | $1^{6}\cdot2^{4}\cdot4^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.1536.97.bq.2 | $56$ | $2$ | $2$ | $97$ | $8$ | $1^{10}\cdot2^{9}\cdot4^{4}\cdot8$ |
| 56.1536.97.bq.4 | $56$ | $2$ | $2$ | $97$ | $8$ | $1^{10}\cdot2^{9}\cdot4^{4}\cdot8$ |
| 56.1536.97.br.1 | $56$ | $2$ | $2$ | $97$ | $10$ | $1^{10}\cdot2^{9}\cdot4^{4}\cdot8$ |
| 56.1536.97.br.3 | $56$ | $2$ | $2$ | $97$ | $10$ | $1^{10}\cdot2^{9}\cdot4^{4}\cdot8$ |
| 56.1536.97.bs.1 | $56$ | $2$ | $2$ | $97$ | $7$ | $1^{10}\cdot2^{9}\cdot4^{4}\cdot8$ |
| 56.1536.97.bs.3 | $56$ | $2$ | $2$ | $97$ | $7$ | $1^{10}\cdot2^{9}\cdot4^{4}\cdot8$ |
| 56.1536.97.bt.2 | $56$ | $2$ | $2$ | $97$ | $3$ | $1^{10}\cdot2^{9}\cdot4^{4}\cdot8$ |
| 56.1536.97.bt.4 | $56$ | $2$ | $2$ | $97$ | $3$ | $1^{10}\cdot2^{9}\cdot4^{4}\cdot8$ |
| 56.1536.105.ci.2 | $56$ | $2$ | $2$ | $105$ | $8$ | $1^{10}\cdot2^{9}\cdot4^{2}\cdot8^{3}$ |
| 56.1536.105.cj.2 | $56$ | $2$ | $2$ | $105$ | $3$ | $1^{10}\cdot2^{9}\cdot4^{2}\cdot8^{3}$ |
| 56.1536.105.cm.2 | $56$ | $2$ | $2$ | $105$ | $10$ | $1^{10}\cdot2^{9}\cdot4^{2}\cdot8^{3}$ |
| 56.1536.105.cn.2 | $56$ | $2$ | $2$ | $105$ | $7$ | $1^{10}\cdot2^{9}\cdot4^{2}\cdot8^{3}$ |
| 56.2304.133.db.2 | $56$ | $3$ | $3$ | $133$ | $1$ | $2^{12}\cdot4^{4}\cdot12^{4}$ |
| 56.2304.133.dn.1 | $56$ | $3$ | $3$ | $133$ | $1$ | $2^{12}\cdot4^{4}\cdot12^{4}$ |
| 56.2304.133.dz.1 | $56$ | $3$ | $3$ | $133$ | $11$ | $1^{20}\cdot2^{2}\cdot4^{4}\cdot6^{4}\cdot12^{2}$ |
| 56.5376.369.g.1 | $56$ | $7$ | $7$ | $369$ | $20$ | $1^{48}\cdot2^{30}\cdot4^{20}\cdot6^{4}\cdot8^{8}\cdot12^{4}$ |