Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
| Index: | $768$ | $\PSL_2$-index: | $768$ | ||||
| Genus: | $49 = 1 + \frac{ 768 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 32 }{2}$ | ||||||
| Cusps: | $32$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{8}\cdot28^{8}\cdot56^{8}$ | Cusp orbits | $2^{8}\cdot4^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $8$ | ||||||
| $\Q$-gonality: | $8 \le \gamma \le 16$ | ||||||
| $\overline{\Q}$-gonality: | $8 \le \gamma \le 16$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.768.49.157 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}17&0\\48&39\end{bmatrix}$, $\begin{bmatrix}17&20\\12&11\end{bmatrix}$, $\begin{bmatrix}23&32\\20&7\end{bmatrix}$, $\begin{bmatrix}39&40\\10&41\end{bmatrix}$, $\begin{bmatrix}47&52\\10&5\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 56.1536.49-56.nw.1.1, 56.1536.49-56.nw.1.2, 56.1536.49-56.nw.1.3, 56.1536.49-56.nw.1.4, 56.1536.49-56.nw.1.5, 56.1536.49-56.nw.1.6, 56.1536.49-56.nw.1.7, 56.1536.49-56.nw.1.8, 56.1536.49-56.nw.1.9, 56.1536.49-56.nw.1.10, 56.1536.49-56.nw.1.11, 56.1536.49-56.nw.1.12, 56.1536.49-56.nw.1.13, 56.1536.49-56.nw.1.14, 56.1536.49-56.nw.1.15, 56.1536.49-56.nw.1.16 |
| Cyclic 56-isogeny field degree: | $2$ |
| Cyclic 56-torsion field degree: | $24$ |
| Full 56-torsion field degree: | $4032$ |
Jacobian
| Conductor: | $2^{196}\cdot7^{75}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{21}\cdot2^{6}\cdot4^{4}$ |
| Newforms: | 14.2.a.a$^{3}$, 56.2.a.a, 56.2.a.b, 56.2.b.a$^{2}$, 56.2.b.b$^{2}$, 98.2.a.a, 392.2.a.b, 392.2.a.d, 392.2.b.b, 392.2.b.c, 448.2.a.a, 448.2.a.c, 448.2.a.d, 448.2.a.e, 448.2.a.g, 448.2.a.h, 784.2.a.b, 784.2.a.e, 784.2.a.i, 1568.2.b.a, 1568.2.b.d, 3136.2.a.bf, 3136.2.a.by, 3136.2.a.f, 3136.2.a.m$^{2}$, 3136.2.a.y |
Rational points
This modular curve has no $\Q_p$ points for $p=5,13$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.96.1.by.1 | $56$ | $8$ | $8$ | $1$ | $1$ | $1^{20}\cdot2^{6}\cdot4^{4}$ |
| 56.384.23.x.1 | $56$ | $2$ | $2$ | $23$ | $2$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.384.23.y.1 | $56$ | $2$ | $2$ | $23$ | $2$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.384.23.df.1 | $56$ | $2$ | $2$ | $23$ | $2$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.384.23.di.1 | $56$ | $2$ | $2$ | $23$ | $2$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.384.25.eh.2 | $56$ | $2$ | $2$ | $25$ | $4$ | $1^{12}\cdot2^{2}\cdot4^{2}$ |
| 56.384.25.ei.1 | $56$ | $2$ | $2$ | $25$ | $4$ | $1^{12}\cdot2^{2}\cdot4^{2}$ |
| 56.384.25.ew.1 | $56$ | $2$ | $2$ | $25$ | $8$ | $2^{4}\cdot4^{4}$ |
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.ii.1 | $56$ | $2$ | $2$ | $97$ | $10$ | $2^{12}\cdot4^{4}\cdot8$ |
| 56.1536.97.ii.3 | $56$ | $2$ | $2$ | $97$ | $10$ | $2^{12}\cdot4^{4}\cdot8$ |
| 56.1536.97.ij.1 | $56$ | $2$ | $2$ | $97$ | $10$ | $2^{12}\cdot4^{4}\cdot8$ |
| 56.1536.97.ij.3 | $56$ | $2$ | $2$ | $97$ | $10$ | $2^{12}\cdot4^{4}\cdot8$ |
| 56.1536.97.jo.1 | $56$ | $2$ | $2$ | $97$ | $10$ | $2^{12}\cdot4^{4}\cdot8$ |
| 56.1536.97.jo.2 | $56$ | $2$ | $2$ | $97$ | $10$ | $2^{12}\cdot4^{4}\cdot8$ |
| 56.1536.97.jp.1 | $56$ | $2$ | $2$ | $97$ | $10$ | $2^{12}\cdot4^{4}\cdot8$ |
| 56.1536.97.jp.2 | $56$ | $2$ | $2$ | $97$ | $10$ | $2^{12}\cdot4^{4}\cdot8$ |
| 56.2304.145.bkm.1 | $56$ | $3$ | $3$ | $145$ | $10$ | $2^{16}\cdot4^{4}\cdot12^{4}$ |
| 56.2304.145.bkm.2 | $56$ | $3$ | $3$ | $145$ | $10$ | $2^{16}\cdot4^{4}\cdot12^{4}$ |
| 56.2304.145.bqa.1 | $56$ | $3$ | $3$ | $145$ | $30$ | $1^{32}\cdot2^{8}\cdot6^{8}$ |
| 56.5376.385.bkq.1 | $56$ | $7$ | $7$ | $385$ | $67$ | $1^{86}\cdot2^{51}\cdot4^{13}\cdot6^{8}\cdot12^{4}$ |