Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $3136$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{8}$ | Cusp orbits | $2^{6}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.192.1.406 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}1&32\\52&49\end{bmatrix}$, $\begin{bmatrix}9&52\\20&3\end{bmatrix}$, $\begin{bmatrix}15&4\\8&19\end{bmatrix}$, $\begin{bmatrix}31&12\\28&15\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.96.1.w.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $16$ |
| Cyclic 56-torsion field degree: | $192$ |
| Full 56-torsion field degree: | $16128$ |
Jacobian
| Conductor: | $2^{6}\cdot7^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 3136.2.a.m |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.0-8.c.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-56.b.1.9 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-56.b.1.23 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-8.c.1.9 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-56.q.1.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-56.q.1.16 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-56.r.1.5 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-56.r.1.12 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.1-56.n.2.9 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.96.1-56.n.2.10 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.96.1-56.bi.2.7 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.96.1-56.bi.2.10 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.96.1-56.bj.2.7 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.96.1-56.bj.2.12 | $56$ | $2$ | $2$ | $1$ | $1$ | 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 |
|---|---|---|---|---|---|---|
| 56.384.5-56.w.1.6 | $56$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 56.384.5-56.y.1.3 | $56$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 56.384.5-56.z.2.7 | $56$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 56.384.5-56.bb.1.3 | $56$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 56.1536.49-56.fs.2.11 | $56$ | $8$ | $8$ | $49$ | $8$ | $1^{20}\cdot2^{6}\cdot4^{4}$ |
| 56.4032.145-56.pe.2.5 | $56$ | $21$ | $21$ | $145$ | $24$ | $1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$ |
| 56.5376.193-56.py.2.6 | $56$ | $28$ | $28$ | $193$ | $31$ | $1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$ |
| 112.384.5-112.c.1.8 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.e.1.8 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.n.1.2 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.t.2.4 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.co.1.4 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.cu.1.4 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.dd.1.6 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.df.1.7 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.hh.1.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.hj.1.11 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.hq.1.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.ht.1.11 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.gz.2.3 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.hb.1.7 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.hi.2.5 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.hl.1.7 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |