Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
| 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: | 8C1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.24.1.92 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}2&39\\47&6\end{bmatrix}$, $\begin{bmatrix}27&18\\50&7\end{bmatrix}$, $\begin{bmatrix}42&27\\45&42\end{bmatrix}$, $\begin{bmatrix}48&33\\47&20\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 112.48.1-56.bh.1.1, 112.48.1-56.bh.1.2, 112.48.1-56.bh.1.3, 112.48.1-56.bh.1.4, 112.48.1-56.bh.1.5, 112.48.1-56.bh.1.6, 112.48.1-56.bh.1.7, 112.48.1-56.bh.1.8 |
| Cyclic 56-isogeny field degree: | $32$ |
| Cyclic 56-torsion field degree: | $768$ |
| Full 56-torsion field degree: | $129024$ |
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 $ | $=$ | $ 14 x y + y w - 2 z w $ |
| $=$ | $56 x^{2} - 4 y^{2} + 2 y z - 2 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 28 x^{4} - 14 x^{3} y + 14 x^{2} y^{2} - 9 x^{2} z^{2} + 8 x y z^{2} - 8 y^{2} z^{2} $ |
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:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^6\,\frac{722064xz^{4}w+122752xz^{2}w^{3}+88396y^{2}z^{4}-1260y^{2}z^{2}w^{2}-5616y^{2}w^{4}-212268yz^{5}-61978yz^{3}w^{2}-2034yzw^{4}+10584z^{6}-13300z^{4}w^{2}+12994z^{2}w^{4}+1323w^{6}}{3136xz^{4}w+2016xz^{2}w^{3}-784y^{2}z^{4}-1456y^{2}z^{2}w^{2}+124y^{2}w^{4}+392yz^{5}+1624yz^{3}w^{2}+290yzw^{4}-392z^{6}-1148z^{4}w^{2}-110z^{2}w^{4}-49w^{6}}$ |
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 |
|---|---|---|---|---|---|---|
| 8.12.1.c.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 28.12.0.n.1 | $28$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.12.0.bo.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.k.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.by.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.di.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.dp.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.fk.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.fq.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.gd.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.gf.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.192.13.cn.1 | $56$ | $8$ | $8$ | $13$ | $6$ | $1^{12}$ |
| 56.504.37.er.1 | $56$ | $21$ | $21$ | $37$ | $9$ | $1^{8}\cdot2^{12}\cdot4$ |
| 56.672.49.er.1 | $56$ | $28$ | $28$ | $49$ | $15$ | $1^{20}\cdot2^{12}\cdot4$ |
| 168.48.1.oz.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.pd.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.qf.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.qj.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.vc.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.vi.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.wl.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.wn.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.5.ez.1 | $168$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 168.96.5.dt.1 | $168$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 280.48.1.od.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.oh.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.pj.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.pn.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.ug.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.um.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.vp.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.vr.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.120.9.cn.1 | $280$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 280.144.9.eb.1 | $280$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 280.240.17.bkd.1 | $280$ | $10$ | $10$ | $17$ | $?$ | not computed |