Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| 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 | $4^{2}$ | ||
| 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: | 8F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.48.1.398 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}22&49\\1&18\end{bmatrix}$, $\begin{bmatrix}29&22\\2&13\end{bmatrix}$, $\begin{bmatrix}49&54\\26&55\end{bmatrix}$ |
| 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: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 y^{2} + y z + z^{2} - w^{2} $ |
| $=$ | $56 x^{2} - 3 y^{2} + 2 y z + 2 z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{4} + 3 x^{2} y^{2} - 112 x^{2} z^{2} + 2 y^{4} - 84 y^{2} z^{2} + 882 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:
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 4x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{7}w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{5294205yz^{11}-15126300yz^{9}w^{2}+20398896yz^{7}w^{4}-15212736yz^{5}w^{6}-17054352yz^{3}w^{8}+12196800yzw^{10}+2705927z^{12}-8521149z^{10}w^{2}-1310946z^{8}w^{4}+21167216z^{6}w^{6}-25565064z^{4}w^{8}+9738960z^{2}w^{10}+5324000w^{12}}{w^{4}(7203yz^{7}-12348yz^{5}w^{2}+15288yz^{3}w^{4}-6048yzw^{6}-16807z^{8}+39445z^{6}w^{2}-25774z^{4}w^{4}+4536z^{2}w^{6}+648w^{8})}$ |
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.be.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.24.0.co.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.24.0.cv.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.24.0.dq.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.24.0.dz.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.24.1.be.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.24.1.bv.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.os.1 | $56$ | $8$ | $8$ | $25$ | $10$ | $1^{20}\cdot2^{2}$ |
| 56.1008.73.brm.1 | $56$ | $21$ | $21$ | $73$ | $41$ | $1^{16}\cdot2^{26}\cdot4$ |
| 56.1344.97.bqs.1 | $56$ | $28$ | $28$ | $97$ | $51$ | $1^{36}\cdot2^{28}\cdot4$ |
| 112.96.3.ln.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.lp.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.op.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.oq.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.or.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.os.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.pf.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.ph.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.5.mf.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.96.5.mg.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.9.ffe.1 | $168$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 168.192.9.bsf.1 | $168$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 280.240.17.bcm.1 | $280$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 280.288.17.dls.1 | $280$ | $6$ | $6$ | $17$ | $?$ | not computed |