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: | $32$ (of which $8$ are rational) | Cusp widths | $4^{8}\cdot8^{8}\cdot28^{8}\cdot56^{8}$ | Cusp orbits | $1^{8}\cdot4^{6}$ | ||
| 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: | $8$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.768.49.129 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}7&12\\22&53\end{bmatrix}$, $\begin{bmatrix}7&16\\10&41\end{bmatrix}$, $\begin{bmatrix}9&16\\42&39\end{bmatrix}$, $\begin{bmatrix}15&28\\48&51\end{bmatrix}$, $\begin{bmatrix}29&16\\46&55\end{bmatrix}$, $\begin{bmatrix}31&0\\32&55\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 56.1536.49-56.fs.1.1, 56.1536.49-56.fs.1.2, 56.1536.49-56.fs.1.3, 56.1536.49-56.fs.1.4, 56.1536.49-56.fs.1.5, 56.1536.49-56.fs.1.6, 56.1536.49-56.fs.1.7, 56.1536.49-56.fs.1.8, 56.1536.49-56.fs.1.9, 56.1536.49-56.fs.1.10, 56.1536.49-56.fs.1.11, 56.1536.49-56.fs.1.12, 56.1536.49-56.fs.1.13, 56.1536.49-56.fs.1.14, 56.1536.49-56.fs.1.15, 56.1536.49-56.fs.1.16, 56.1536.49-56.fs.1.17, 56.1536.49-56.fs.1.18, 56.1536.49-56.fs.1.19, 56.1536.49-56.fs.1.20, 56.1536.49-56.fs.1.21, 56.1536.49-56.fs.1.22, 56.1536.49-56.fs.1.23, 56.1536.49-56.fs.1.24 |
| 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$^{4}$, 56.2.a.a$^{2}$, 56.2.a.b$^{2}$, 56.2.b.a$^{2}$, 56.2.b.b$^{2}$, 112.2.a.a, 112.2.a.b, 112.2.a.c, 392.2.b.b, 392.2.b.c, 1568.2.b.a, 1568.2.b.d, 3136.2.a.bf, 3136.2.a.by, 3136.2.a.c, 3136.2.a.e, 3136.2.a.f, 3136.2.a.m$^{2}$, 3136.2.a.p, 3136.2.a.q, 3136.2.a.w, 3136.2.a.y, 3136.2.a.z |
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 |
|---|---|---|---|---|---|---|
| 56.96.1.w.2 | $56$ | $8$ | $8$ | $1$ | $1$ | $1^{20}\cdot2^{6}\cdot4^{4}$ |
| 56.384.23.f.1 | $56$ | $2$ | $2$ | $23$ | $1$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.384.23.i.1 | $56$ | $2$ | $2$ | $23$ | $1$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.384.23.cr.1 | $56$ | $2$ | $2$ | $23$ | $3$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.384.23.cs.1 | $56$ | $2$ | $2$ | $23$ | $3$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.384.25.bj.1 | $56$ | $2$ | $2$ | $25$ | $8$ | $2^{4}\cdot4^{4}$ |
| 56.384.25.eh.2 | $56$ | $2$ | $2$ | $25$ | $4$ | $1^{12}\cdot2^{2}\cdot4^{2}$ |
| 56.384.25.ei.2 | $56$ | $2$ | $2$ | $25$ | $4$ | $1^{12}\cdot2^{2}\cdot4^{2}$ |
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.br.2 | $56$ | $2$ | $2$ | $97$ | $10$ | $2^{12}\cdot4^{4}\cdot8$ |
| 56.1536.97.br.3 | $56$ | $2$ | $2$ | $97$ | $10$ | $2^{12}\cdot4^{4}\cdot8$ |
| 56.1536.97.bv.2 | $56$ | $2$ | $2$ | $97$ | $10$ | $2^{12}\cdot4^{4}\cdot8$ |
| 56.1536.97.bv.3 | $56$ | $2$ | $2$ | $97$ | $10$ | $2^{12}\cdot4^{4}\cdot8$ |
| 56.1536.97.fv.1 | $56$ | $2$ | $2$ | $97$ | $10$ | $2^{12}\cdot4^{4}\cdot8$ |
| 56.1536.97.fv.2 | $56$ | $2$ | $2$ | $97$ | $10$ | $2^{12}\cdot4^{4}\cdot8$ |
| 56.1536.97.fz.1 | $56$ | $2$ | $2$ | $97$ | $10$ | $2^{12}\cdot4^{4}\cdot8$ |
| 56.1536.97.fz.2 | $56$ | $2$ | $2$ | $97$ | $10$ | $2^{12}\cdot4^{4}\cdot8$ |
| 56.1536.105.fz.1 | $56$ | $2$ | $2$ | $105$ | $20$ | $1^{20}\cdot2^{10}\cdot4^{4}$ |
| 56.1536.105.ge.2 | $56$ | $2$ | $2$ | $105$ | $16$ | $1^{20}\cdot2^{10}\cdot4^{4}$ |
| 56.1536.105.gg.2 | $56$ | $2$ | $2$ | $105$ | $20$ | $1^{20}\cdot2^{10}\cdot4^{4}$ |
| 56.1536.105.gl.4 | $56$ | $2$ | $2$ | $105$ | $16$ | $1^{20}\cdot2^{10}\cdot4^{4}$ |
| 56.1536.105.gn.1 | $56$ | $2$ | $2$ | $105$ | $8$ | $2^{8}\cdot4^{2}\cdot8^{4}$ |
| 56.1536.105.gn.2 | $56$ | $2$ | $2$ | $105$ | $8$ | $2^{8}\cdot4^{2}\cdot8^{4}$ |
| 56.1536.105.gp.1 | $56$ | $2$ | $2$ | $105$ | $12$ | $2^{8}\cdot4^{2}\cdot8^{4}$ |
| 56.1536.105.gp.2 | $56$ | $2$ | $2$ | $105$ | $12$ | $2^{8}\cdot4^{2}\cdot8^{4}$ |
| 56.2304.145.na.1 | $56$ | $3$ | $3$ | $145$ | $10$ | $2^{16}\cdot4^{4}\cdot12^{4}$ |
| 56.2304.145.na.2 | $56$ | $3$ | $3$ | $145$ | $10$ | $2^{16}\cdot4^{4}\cdot12^{4}$ |
| 56.2304.145.ts.1 | $56$ | $3$ | $3$ | $145$ | $30$ | $1^{32}\cdot2^{8}\cdot6^{8}$ |
| 56.5376.385.jm.1 | $56$ | $7$ | $7$ | $385$ | $67$ | $1^{86}\cdot2^{51}\cdot4^{13}\cdot6^{8}\cdot12^{4}$ |