Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $1568$ | ||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $2^{2}$ | ||
| 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: | 8B1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.24.1.59 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}4&51\\45&4\end{bmatrix}$, $\begin{bmatrix}35&16\\24&21\end{bmatrix}$, $\begin{bmatrix}44&37\\41&36\end{bmatrix}$, $\begin{bmatrix}46&3\\43&38\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 56-isogeny field degree: | $32$ |
| Cyclic 56-torsion field degree: | $768$ |
| Full 56-torsion field degree: | $129024$ |
Jacobian
| Conductor: | $2^{5}\cdot7^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 1568.2.a.e |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 4 x^{2} + z w $ |
| $=$ | $14 y^{2} - 4 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 14 y^{2} z^{2} - 4 z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{4}w$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^8\,\frac{(z-w)^{3}(z+w)^{3}}{w^{2}z^{4}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.12.0.z.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.12.0.bn.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.12.1.d.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 |
|---|---|---|---|---|---|---|
| 56.48.1.ii.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.48.1.ij.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.48.1.ik.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.48.1.il.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.48.1.jw.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.48.1.jx.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.48.1.jy.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.48.1.jz.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.192.13.ff.1 | $56$ | $8$ | $8$ | $13$ | $6$ | $1^{8}\cdot2^{2}$ |
| 56.504.37.ot.1 | $56$ | $21$ | $21$ | $37$ | $22$ | $1^{4}\cdot2^{14}\cdot4$ |
| 56.672.49.ot.1 | $56$ | $28$ | $28$ | $49$ | $27$ | $1^{12}\cdot2^{16}\cdot4$ |
| 112.48.3.bo.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.48.3.bo.2 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.48.3.ea.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.48.3.ea.2 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.48.3.ec.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.48.3.ec.2 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.48.3.gg.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.48.3.gg.2 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.48.1.clk.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.cll.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.clm.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.cln.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.cmq.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.cmr.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.cms.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.cmt.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.5.brz.1 | $168$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 168.96.5.uv.1 | $168$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 280.48.1.cfq.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.cfr.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.cfs.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.cft.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.cgw.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.cgx.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.cgy.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.cgz.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.120.9.ol.1 | $280$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 280.144.9.bnj.1 | $280$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 280.240.17.fhx.1 | $280$ | $10$ | $10$ | $17$ | $?$ | not computed |