Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
| Index: | $1536$ | $\PSL_2$-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: | 56.1536.49.15153 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}1&36\\32&5\end{bmatrix}$, $\begin{bmatrix}25&20\\34&11\end{bmatrix}$, $\begin{bmatrix}39&36\\6&13\end{bmatrix}$, $\begin{bmatrix}55&48\\2&31\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.768.49.nw.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $2$ |
| Cyclic 56-torsion field degree: | $24$ |
| Full 56-torsion field degree: | $2016$ |
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 |
|---|---|---|---|---|---|---|
| 56.768.23-56.x.1.8 | $56$ | $2$ | $2$ | $23$ | $2$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.768.23-56.x.1.24 | $56$ | $2$ | $2$ | $23$ | $2$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.768.23-56.y.1.8 | $56$ | $2$ | $2$ | $23$ | $2$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.768.23-56.y.1.23 | $56$ | $2$ | $2$ | $23$ | $2$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.768.23-56.df.1.13 | $56$ | $2$ | $2$ | $23$ | $2$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.768.23-56.df.1.28 | $56$ | $2$ | $2$ | $23$ | $2$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.768.23-56.di.1.3 | $56$ | $2$ | $2$ | $23$ | $2$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.768.23-56.di.1.24 | $56$ | $2$ | $2$ | $23$ | $2$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.768.25-56.eh.2.12 | $56$ | $2$ | $2$ | $25$ | $4$ | $1^{12}\cdot2^{2}\cdot4^{2}$ |
| 56.768.25-56.eh.2.29 | $56$ | $2$ | $2$ | $25$ | $4$ | $1^{12}\cdot2^{2}\cdot4^{2}$ |
| 56.768.25-56.ei.1.4 | $56$ | $2$ | $2$ | $25$ | $4$ | $1^{12}\cdot2^{2}\cdot4^{2}$ |
| 56.768.25-56.ei.1.22 | $56$ | $2$ | $2$ | $25$ | $4$ | $1^{12}\cdot2^{2}\cdot4^{2}$ |
| 56.768.25-56.ew.1.15 | $56$ | $2$ | $2$ | $25$ | $8$ | $2^{4}\cdot4^{4}$ |
| 56.768.25-56.ew.1.16 | $56$ | $2$ | $2$ | $25$ | $8$ | $2^{4}\cdot4^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.3072.97-56.ii.1.8 | $56$ | $2$ | $2$ | $97$ | $10$ | $2^{12}\cdot4^{4}\cdot8$ |
| 56.3072.97-56.ii.3.8 | $56$ | $2$ | $2$ | $97$ | $10$ | $2^{12}\cdot4^{4}\cdot8$ |
| 56.3072.97-56.ij.1.4 | $56$ | $2$ | $2$ | $97$ | $10$ | $2^{12}\cdot4^{4}\cdot8$ |
| 56.3072.97-56.ij.3.4 | $56$ | $2$ | $2$ | $97$ | $10$ | $2^{12}\cdot4^{4}\cdot8$ |
| 56.3072.97-56.jo.1.2 | $56$ | $2$ | $2$ | $97$ | $10$ | $2^{12}\cdot4^{4}\cdot8$ |
| 56.3072.97-56.jo.2.2 | $56$ | $2$ | $2$ | $97$ | $10$ | $2^{12}\cdot4^{4}\cdot8$ |
| 56.3072.97-56.jp.1.4 | $56$ | $2$ | $2$ | $97$ | $10$ | $2^{12}\cdot4^{4}\cdot8$ |
| 56.3072.97-56.jp.2.4 | $56$ | $2$ | $2$ | $97$ | $10$ | $2^{12}\cdot4^{4}\cdot8$ |
| 56.4608.145-56.bkm.1.13 | $56$ | $3$ | $3$ | $145$ | $10$ | $2^{16}\cdot4^{4}\cdot12^{4}$ |
| 56.4608.145-56.bkm.2.13 | $56$ | $3$ | $3$ | $145$ | $10$ | $2^{16}\cdot4^{4}\cdot12^{4}$ |
| 56.4608.145-56.bqa.1.8 | $56$ | $3$ | $3$ | $145$ | $30$ | $1^{32}\cdot2^{8}\cdot6^{8}$ |
| 56.10752.385-56.bkq.1.5 | $56$ | $7$ | $7$ | $385$ | $67$ | $1^{86}\cdot2^{51}\cdot4^{13}\cdot6^{8}\cdot12^{4}$ |