Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $1568$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $2$ are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $1^{2}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.96.1.39 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}19&40\\50&9\end{bmatrix}$, $\begin{bmatrix}35&4\\26&29\end{bmatrix}$, $\begin{bmatrix}37&4\\12&29\end{bmatrix}$, $\begin{bmatrix}53&16\\36&47\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.1.bf.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $16$ |
| Cyclic 56-torsion field degree: | $192$ |
| Full 56-torsion field degree: | $32256$ |
Jacobian
| Conductor: | $2^{5}\cdot7^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 1568.2.a.e |
Rational points
This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.0-8.e.2.13 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.48.0-8.e.2.6 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.48.0-56.i.1.14 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.48.0-56.i.1.24 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.48.1-56.c.1.2 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.48.1-56.c.1.12 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.192.1-56.s.1.2 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.192.1-56.x.2.5 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.192.1-56.bf.2.2 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.192.1-56.bh.1.3 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.192.1-56.bx.2.5 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.192.1-56.bz.1.2 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.192.1-56.cc.1.4 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.192.1-56.cd.2.6 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.768.25-56.ea.2.27 | $56$ | $8$ | $8$ | $25$ | $4$ | $1^{8}\cdot2^{4}\cdot4^{2}$ |
| 56.2016.73-56.gx.1.26 | $56$ | $21$ | $21$ | $73$ | $10$ | $1^{4}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.2688.97-56.gx.2.29 | $56$ | $28$ | $28$ | $97$ | $13$ | $1^{12}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 168.192.1-168.ga.2.9 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.ge.2.3 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.hh.2.14 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.hl.2.9 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.mj.2.9 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.mn.2.15 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.np.2.5 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.nt.2.9 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.288.9-168.rv.1.55 | $168$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 168.384.9-168.jl.1.54 | $168$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 280.192.1-280.ga.2.3 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.ge.2.10 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.hh.2.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.hl.2.7 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.lp.2.13 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.lt.2.3 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.mv.2.11 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.mz.2.5 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.480.17-280.ep.1.27 | $280$ | $5$ | $5$ | $17$ | $?$ | not computed |