Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| 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 | $4^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.96.1.1086 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}29&46\\34&53\end{bmatrix}$, $\begin{bmatrix}31&22\\40&29\end{bmatrix}$, $\begin{bmatrix}31&26\\22&47\end{bmatrix}$, $\begin{bmatrix}51&0\\50&53\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 56.192.1-56.k.1.1, 56.192.1-56.k.1.2, 56.192.1-56.k.1.3, 56.192.1-56.k.1.4, 56.192.1-56.k.1.5, 56.192.1-56.k.1.6, 56.192.1-56.k.1.7, 56.192.1-56.k.1.8, 112.192.1-56.k.1.1, 112.192.1-56.k.1.2, 112.192.1-56.k.1.3, 112.192.1-56.k.1.4, 168.192.1-56.k.1.1, 168.192.1-56.k.1.2, 168.192.1-56.k.1.3, 168.192.1-56.k.1.4, 168.192.1-56.k.1.5, 168.192.1-56.k.1.6, 168.192.1-56.k.1.7, 168.192.1-56.k.1.8, 280.192.1-56.k.1.1, 280.192.1-56.k.1.2, 280.192.1-56.k.1.3, 280.192.1-56.k.1.4, 280.192.1-56.k.1.5, 280.192.1-56.k.1.6, 280.192.1-56.k.1.7, 280.192.1-56.k.1.8 |
| Cyclic 56-isogeny field degree: | $16$ |
| Cyclic 56-torsion field degree: | $384$ |
| Full 56-torsion field degree: | $32256$ |
Jacobian
| Conductor: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 14 x^{2} - 7 y^{2} - w^{2} $ |
| $=$ | $14 y^{2} - 2 z^{2} + w^{2}$ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^2\,\frac{(16z^{8}+56z^{4}w^{4}+w^{8})^{3}}{w^{4}z^{4}(2z^{2}-w^{2})^{4}(2z^{2}+w^{2})^{4}}$ |
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 |
|---|---|---|---|---|---|---|
| 8.48.1.j.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.0.d.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.48.0.e.2 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.48.0.w.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.48.0.x.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.48.1.m.2 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.q.1 | $56$ | $2$ | $2$ | $1$ | $0$ | 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.dz.1 | $56$ | $8$ | $8$ | $49$ | $5$ | $1^{20}\cdot2^{6}\cdot4^{4}$ |
| 56.2016.145.ll.2 | $56$ | $21$ | $21$ | $145$ | $27$ | $1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$ |
| 56.2688.193.mf.1 | $56$ | $28$ | $28$ | $193$ | $32$ | $1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$ |
| 112.192.9.dv.1 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.192.9.dy.1 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.192.9.eb.1 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.192.9.ec.1 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.288.17.bis.2 | $168$ | $3$ | $3$ | $17$ | $?$ | not computed |
| 168.384.17.mq.2 | $168$ | $4$ | $4$ | $17$ | $?$ | not computed |