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: | 8B1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.24.1.93 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}1&42\\42&9\end{bmatrix}$, $\begin{bmatrix}7&46\\10&1\end{bmatrix}$, $\begin{bmatrix}12&29\\13&40\end{bmatrix}$, $\begin{bmatrix}46&3\\49&54\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 112.48.1-56.y.1.1, 112.48.1-56.y.1.2, 112.48.1-56.y.1.3, 112.48.1-56.y.1.4, 112.48.1-56.y.1.5, 112.48.1-56.y.1.6, 112.48.1-56.y.1.7, 112.48.1-56.y.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 $ | $=$ | $ 2 y^{2} - y z + z^{2} - w^{2} $ |
| $=$ | $28 x^{2} - y w + 2 z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{4} + 7 x^{2} y z + 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{14}w$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{1715yz^{5}-196yz^{3}w^{2}+84yzw^{4}+343z^{6}-1421z^{4}w^{2}+210z^{2}w^{4}+4w^{6}}{w^{4}(7yz-7z^{2}-w^{2})}$ |
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.c.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 28.12.0.k.1 | $28$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.12.0.bx.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.eu.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.ev.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.ew.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.ex.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.ga.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.gb.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.gc.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.gd.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.192.13.bu.1 | $56$ | $8$ | $8$ | $13$ | $3$ | $1^{8}\cdot2^{2}$ |
| 56.504.37.cu.1 | $56$ | $21$ | $21$ | $37$ | $10$ | $1^{4}\cdot2^{14}\cdot4$ |
| 56.672.49.cu.1 | $56$ | $28$ | $28$ | $49$ | $13$ | $1^{12}\cdot2^{16}\cdot4$ |
| 168.48.1.rk.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.rl.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.rm.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.rn.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.sq.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.sr.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.ss.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.st.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.5.cu.1 | $168$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 168.96.5.cs.1 | $168$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 280.48.1.qo.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.qp.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.qq.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.qr.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.ru.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.rv.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.rw.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.rx.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.120.9.bo.1 | $280$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 280.144.9.cm.1 | $280$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 280.240.17.bhy.1 | $280$ | $10$ | $10$ | $17$ | $?$ | not computed |