Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| 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: | $0$ | ||||||
| $\Q$-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: | 56.24.1.82 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}12&25\\51&12\end{bmatrix}$, $\begin{bmatrix}35&52\\20&5\end{bmatrix}$, $\begin{bmatrix}39&18\\10&19\end{bmatrix}$, $\begin{bmatrix}48&3\\53&4\end{bmatrix}$ |
| 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: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 7 y^{2} + 4 z^{2} - w^{2} $ |
| $=$ | $14 x^{2} - z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 7 y^{2} z^{2} - 49 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{14}w$ |
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 |
|---|---|---|---|---|---|---|
| 8.12.1.d.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 28.12.0.l.1 | $28$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.12.0.bw.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
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.fs.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.ft.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.fu.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.fv.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.gy.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.gz.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.ha.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.hb.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.192.13.ca.1 | $56$ | $8$ | $8$ | $13$ | $5$ | $1^{8}\cdot2^{2}$ |
| 56.504.37.da.1 | $56$ | $21$ | $21$ | $37$ | $14$ | $1^{4}\cdot2^{14}\cdot4$ |
| 56.672.49.da.1 | $56$ | $28$ | $28$ | $49$ | $19$ | $1^{12}\cdot2^{16}\cdot4$ |
| 168.48.1.si.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.sj.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.sk.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.sl.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.to.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.tp.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.tq.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.tr.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.5.da.1 | $168$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 168.96.5.cy.1 | $168$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 280.48.1.rm.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.rn.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.ro.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.rp.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.ss.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.st.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.su.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.sv.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.120.9.bu.1 | $280$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 280.144.9.cs.1 | $280$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 280.240.17.bie.1 | $280$ | $10$ | $10$ | $17$ | $?$ | not computed |