Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $3136$ | ||
| Index: | $96$ | $\PSL_2$-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 | $2^{4}\cdot4^{2}$ | ||
| 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: | 56.96.1.383 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}3&34\\12&21\end{bmatrix}$, $\begin{bmatrix}25&10\\24&51\end{bmatrix}$, $\begin{bmatrix}47&14\\50&15\end{bmatrix}$, $\begin{bmatrix}49&50\\32&7\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 56.192.1-56.by.1.1, 56.192.1-56.by.1.2, 56.192.1-56.by.1.3, 56.192.1-56.by.1.4, 56.192.1-56.by.1.5, 56.192.1-56.by.1.6, 56.192.1-56.by.1.7, 56.192.1-56.by.1.8, 168.192.1-56.by.1.1, 168.192.1-56.by.1.2, 168.192.1-56.by.1.3, 168.192.1-56.by.1.4, 168.192.1-56.by.1.5, 168.192.1-56.by.1.6, 168.192.1-56.by.1.7, 168.192.1-56.by.1.8, 280.192.1-56.by.1.1, 280.192.1-56.by.1.2, 280.192.1-56.by.1.3, 280.192.1-56.by.1.4, 280.192.1-56.by.1.5, 280.192.1-56.by.1.6, 280.192.1-56.by.1.7, 280.192.1-56.by.1.8 |
| Cyclic 56-isogeny field degree: | $16$ |
| Cyclic 56-torsion field degree: | $192$ |
| Full 56-torsion field degree: | $32256$ |
Jacobian
| Conductor: | $2^{6}\cdot7^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 3136.2.a.m |
Rational points
This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.0.i.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.48.0.i.2 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.48.0.j.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.48.0.x.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.48.1.bi.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.48.1.bj.2 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.48.1.bs.1 | $56$ | $2$ | $2$ | $1$ | $1$ | 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 |
|---|---|---|---|---|---|---|
| 56.768.49.nw.1 | $56$ | $8$ | $8$ | $49$ | $8$ | $1^{20}\cdot2^{6}\cdot4^{4}$ |
| 56.2016.145.bks.2 | $56$ | $21$ | $21$ | $145$ | $24$ | $1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$ |
| 56.2688.193.blu.1 | $56$ | $28$ | $28$ | $193$ | $31$ | $1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$ |
| 168.288.17.jcw.1 | $168$ | $3$ | $3$ | $17$ | $?$ | not computed |
| 168.384.17.drw.2 | $168$ | $4$ | $4$ | $17$ | $?$ | not computed |