Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $3136$ | ||
| Index: | $192$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (none of which are rational) | Cusp widths | $8^{24}$ | Cusp orbits | $2^{4}\cdot4^{2}\cdot8$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8A5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.192.5.107 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}3&28\\52&51\end{bmatrix}$, $\begin{bmatrix}23&54\\12&41\end{bmatrix}$, $\begin{bmatrix}33&34\\28&47\end{bmatrix}$, $\begin{bmatrix}47&34\\12&53\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 56.384.5-56.f.3.1, 56.384.5-56.f.3.2, 56.384.5-56.f.3.3, 56.384.5-56.f.3.4, 56.384.5-56.f.3.5, 56.384.5-56.f.3.6, 56.384.5-56.f.3.7, 56.384.5-56.f.3.8, 112.384.5-56.f.3.1, 112.384.5-56.f.3.2, 112.384.5-56.f.3.3, 112.384.5-56.f.3.4, 168.384.5-56.f.3.1, 168.384.5-56.f.3.2, 168.384.5-56.f.3.3, 168.384.5-56.f.3.4, 168.384.5-56.f.3.5, 168.384.5-56.f.3.6, 168.384.5-56.f.3.7, 168.384.5-56.f.3.8, 280.384.5-56.f.3.1, 280.384.5-56.f.3.2, 280.384.5-56.f.3.3, 280.384.5-56.f.3.4, 280.384.5-56.f.3.5, 280.384.5-56.f.3.6, 280.384.5-56.f.3.7, 280.384.5-56.f.3.8 |
| Cyclic 56-isogeny field degree: | $8$ |
| Cyclic 56-torsion field degree: | $192$ |
| Full 56-torsion field degree: | $16128$ |
Jacobian
| Conductor: | $2^{28}\cdot7^{8}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 32.2.a.a, 1568.2.a.e, 3136.2.a.m, 3136.2.b.b |
Rational points
This modular curve has no real points, and therefore no rational points.
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 |
|---|---|---|---|---|---|---|
| 8.96.1.b.2 | $8$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 56.96.1.b.2 | $56$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 56.96.1.n.2 | $56$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 56.96.3.d.1 | $56$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 56.96.3.o.3 | $56$ | $2$ | $2$ | $3$ | $2$ | $2$ |
| 56.96.3.p.2 | $56$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 56.96.3.r.2 | $56$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.1536.105.dd.2 | $56$ | $8$ | $8$ | $105$ | $16$ | $1^{38}\cdot2^{15}\cdot4^{8}$ |
| 56.4032.301.jd.4 | $56$ | $21$ | $21$ | $301$ | $51$ | $1^{28}\cdot2^{54}\cdot4^{6}\cdot6^{6}\cdot12^{7}\cdot16$ |
| 56.5376.401.kr.2 | $56$ | $28$ | $28$ | $401$ | $65$ | $1^{66}\cdot2^{69}\cdot4^{14}\cdot6^{6}\cdot12^{7}\cdot16$ |
| 112.384.17.cl.5 | $112$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 112.384.17.cl.6 | $112$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 112.384.17.cm.5 | $112$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 112.384.17.cm.6 | $112$ | $2$ | $2$ | $17$ | $?$ | not computed |