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: | 56.48.1.377 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}0&27\\55&28\end{bmatrix}$, $\begin{bmatrix}3&6\\42&17\end{bmatrix}$, $\begin{bmatrix}35&46\\54&35\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 112.96.1-56.ex.1.1, 112.96.1-56.ex.1.2, 112.96.1-56.ex.1.3, 112.96.1-56.ex.1.4 |
| Cyclic 56-isogeny field degree: | $16$ |
| Cyclic 56-torsion field degree: | $384$ |
| 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 z - 2 z^{2} - 2 w^{2} $ |
| $=$ | $56 x^{2} - 2 y^{2} - y z - z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} - 16 x^{2} y^{2} + 42 x^{2} z^{2} + 9 y^{4} - 84 y^{2} z^{2} + 196 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:
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 4x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{2}{7}w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^6}{3^4}\cdot\frac{1915325720yz^{11}+4104269400yz^{9}w^{2}+2414676096yz^{7}w^{4}+89201952yz^{5}w^{6}-108718848yz^{3}w^{8}+16329600yzw^{10}+1050958517z^{12}+2976082718z^{10}w^{2}+2719103688z^{8}w^{4}+692006616z^{6}w^{6}-109438560z^{4}w^{8}-4626720z^{2}w^{10}-1944000w^{12}}{w^{4}(883568yz^{7}+1136016yz^{5}w^{2}+437472yz^{3}w^{4}+48384yzw^{6}+487403z^{8}+961772z^{6}w^{2}+602700z^{4}w^{4}+133056z^{2}w^{6}+6912w^{8})}$ |
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.t.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.24.0.bx.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.24.0.cf.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.24.0.df.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.24.0.dt.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.24.1.y.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.24.1.bp.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.mj.1 | $56$ | $8$ | $8$ | $25$ | $8$ | $1^{20}\cdot2^{2}$ |
| 56.1008.73.bmx.1 | $56$ | $21$ | $21$ | $73$ | $17$ | $1^{16}\cdot2^{26}\cdot4$ |
| 56.1344.97.bmd.1 | $56$ | $28$ | $28$ | $97$ | $25$ | $1^{36}\cdot2^{28}\cdot4$ |
| 168.144.9.evr.1 | $168$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 168.192.9.bni.1 | $168$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 280.240.17.xx.1 | $280$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 280.288.17.der.1 | $280$ | $6$ | $6$ | $17$ | $?$ | not computed |