Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
| 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: | $0$ | ||||||
| $\Q$-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: | 56.48.1.359 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}15&0\\32&33\end{bmatrix}$, $\begin{bmatrix}44&7\\3&16\end{bmatrix}$, $\begin{bmatrix}47&24\\24&51\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 112.96.1-56.fl.1.1, 112.96.1-56.fl.1.2, 112.96.1-56.fl.1.3, 112.96.1-56.fl.1.4 |
| Cyclic 56-isogeny field degree: | $32$ |
| Cyclic 56-torsion field degree: | $768$ |
| Full 56-torsion field degree: | $64512$ |
Jacobian
| Conductor: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 y^{2} + 2 y w - 2 z^{2} - 2 w^{2} $ |
| $=$ | $14 x^{2} - 2 y^{2} + y w - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 196 x^{4} - 84 x^{2} y^{2} + 42 x^{2} z^{2} + 9 y^{4} - 16 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:
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 7x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{7}{2}w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^6}{3^8}\cdot\frac{493970400yz^{10}w+10685310048yz^{8}w^{3}+71961378048yz^{6}w^{5}+206068955904yz^{4}w^{7}+262673241600yz^{2}w^{9}+122580846080yw^{11}-40429125z^{12}-2123324280z^{10}w^{2}-23226614460z^{8}w^{4}-100378966464z^{6}w^{6}-202447978992z^{4}w^{8}-190469293952z^{2}w^{10}-67261345088w^{12}}{z^{8}(168yz^{2}w+392yw^{3}-48z^{4}-266z^{2}w^{2}-245w^{4})}$ |
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 |
|---|---|---|---|---|---|---|
| 8.24.1.r.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 28.24.0.i.1 | $28$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.24.0.ca.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.24.0.dc.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.24.0.dl.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.24.1.bc.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.24.1.bg.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 |
|---|---|---|---|---|---|---|
| 56.384.25.mx.1 | $56$ | $8$ | $8$ | $25$ | $8$ | $1^{20}\cdot2^{2}$ |
| 56.1008.73.bnl.1 | $56$ | $21$ | $21$ | $73$ | $17$ | $1^{16}\cdot2^{26}\cdot4$ |
| 56.1344.97.bmr.1 | $56$ | $28$ | $28$ | $97$ | $25$ | $1^{36}\cdot2^{28}\cdot4$ |
| 168.144.9.ewf.1 | $168$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 168.192.9.bnw.1 | $168$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 280.240.17.yl.1 | $280$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 280.288.17.dff.1 | $280$ | $6$ | $6$ | $17$ | $?$ | not computed |