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: | 8F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.96.1.131 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}15&0\\50&29\end{bmatrix}$, $\begin{bmatrix}29&2\\0&31\end{bmatrix}$, $\begin{bmatrix}43&4\\18&13\end{bmatrix}$, $\begin{bmatrix}53&32\\28&25\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.1.bu.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $8$ |
| 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.i.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 28.48.0-28.c.1.3 | $28$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.48.0-28.c.1.9 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.48.0-8.i.1.8 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.48.1-56.c.1.6 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.48.1-56.c.1.17 | $56$ | $2$ | $2$ | $1$ | $1$ | 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.ca.1.3 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.192.1-56.ca.2.7 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.192.1-56.cb.1.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.192.1-56.cb.2.2 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.192.1-56.cc.1.2 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.192.1-56.cc.2.4 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.192.1-56.cd.1.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.192.1-56.cd.2.3 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.768.25-56.ey.1.21 | $56$ | $8$ | $8$ | $25$ | $7$ | $1^{20}\cdot2^{2}$ |
| 56.2016.73-56.je.1.12 | $56$ | $21$ | $21$ | $73$ | $20$ | $1^{16}\cdot2^{26}\cdot4$ |
| 56.2688.97-56.je.1.3 | $56$ | $28$ | $28$ | $97$ | $26$ | $1^{36}\cdot2^{28}\cdot4$ |
| 112.192.3-112.bo.1.6 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.bo.2.6 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.cb.1.5 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.cb.2.5 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.ci.1.5 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.ci.2.5 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.cu.1.6 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.cu.2.6 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.1-168.os.1.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.os.2.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.ot.1.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.ot.2.3 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.ou.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.ou.2.13 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.ov.1.3 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.ov.2.14 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.288.9-168.bbe.1.57 | $168$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 168.384.9-168.oi.1.58 | $168$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 280.192.1-280.ny.1.7 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.ny.2.4 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.nz.1.8 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.nz.2.3 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.oa.1.4 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.oa.2.3 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.ob.1.3 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.ob.2.4 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.480.17-280.he.1.26 | $280$ | $5$ | $5$ | $17$ | $?$ | not computed |