Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $1568$ | ||
| Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $2^{2}\cdot4$ | ||
| 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: | 8F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.48.1.174 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}2&5\\11&14\end{bmatrix}$, $\begin{bmatrix}18&47\\9&26\end{bmatrix}$, $\begin{bmatrix}26&45\\53&34\end{bmatrix}$ |
| 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: | $64512$ |
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} - 2 x y + 2 y^{2} - z^{2} $ |
| $=$ | $3 x^{2} + 2 x y - 2 y^{2} + 3 z^{2} - 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 225 x^{4} - 364 x^{2} y^{2} - 60 x^{2} z^{2} + 196 y^{4} + 42 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:
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{(z^{2}+w^{2})^{3}(3z^{2}-w^{2})^{3}}{z^{4}(z-w)^{4}(z+w)^{4}}$ |
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 |
|---|---|---|---|---|---|---|
| 8.24.0.v.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.24.0.w.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.24.0.dl.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.24.0.dm.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.24.1.p.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.24.1.ci.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.24.1.cl.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 |
|---|---|---|---|---|---|---|
| 56.384.25.jr.1 | $56$ | $8$ | $8$ | $25$ | $8$ | $1^{20}\cdot2^{2}$ |
| 56.1008.73.bab.1 | $56$ | $21$ | $21$ | $73$ | $33$ | $1^{16}\cdot2^{26}\cdot4$ |
| 56.1344.97.zl.1 | $56$ | $28$ | $28$ | $97$ | $40$ | $1^{36}\cdot2^{28}\cdot4$ |
| 112.96.3.gz.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.ha.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.hb.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.hc.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.9.cut.1 | $168$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 168.192.9.bbi.1 | $168$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 280.240.17.pn.1 | $280$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 280.288.17.bxj.1 | $280$ | $6$ | $6$ | $17$ | $?$ | not computed |