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: | 56.48.1.380 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}4&21\\23&8\end{bmatrix}$, $\begin{bmatrix}37&26\\34&13\end{bmatrix}$, $\begin{bmatrix}41&14\\50&15\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 112.96.1-56.fq.1.1, 112.96.1-56.fq.1.2, 112.96.1-56.fq.1.3, 112.96.1-56.fq.1.4, 112.96.1-56.fq.1.5, 112.96.1-56.fq.1.6 |
| 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 $ | $=$ | $ 9 y^{2} + 8 y z + 8 z^{2} - 2 w^{2} $ |
| $=$ | $28 x^{2} - 2 y^{2} - y z - z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{4} + 20 x^{2} y^{2} - 21 x^{2} z^{2} + 162 y^{4} - 252 y^{2} z^{2} + 98 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:
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{2}{7}w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^8\cdot3^3\,\frac{3196523330yz^{11}+15868698804yz^{9}w^{2}-1673314524yz^{7}w^{4}-2632494816yz^{5}w^{6}+782504226yz^{3}w^{8}-124002900yzw^{10}-4997847169z^{12}+10191260590z^{10}w^{2}+7314624891z^{8}w^{4}-4480064316z^{6}w^{6}+354102273z^{4}w^{8}-17911530z^{2}w^{10}+7381125w^{12}}{12786093320yz^{11}-62952635340yz^{9}w^{2}+56675653020yz^{7}w^{4}-16417085832yz^{5}w^{6}+2643925536yz^{3}w^{8}-317447424yzw^{10}-19991388676z^{12}+5494746124z^{10}w^{2}+25725752199z^{8}w^{4}-17209123596z^{6}w^{6}+3954886236z^{4}w^{8}-379173312z^{2}w^{10}-15116544w^{12}}$ |
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.s.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.24.0.cd.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.24.0.cr.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.24.0.db.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.24.0.dg.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.24.1.bd.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.24.1.bh.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.nc.1 | $56$ | $8$ | $8$ | $25$ | $10$ | $1^{20}\cdot2^{2}$ |
| 56.1008.73.bnq.1 | $56$ | $21$ | $21$ | $73$ | $28$ | $1^{16}\cdot2^{26}\cdot4$ |
| 56.1344.97.bmw.1 | $56$ | $28$ | $28$ | $97$ | $38$ | $1^{36}\cdot2^{28}\cdot4$ |
| 112.96.3.mp.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.mq.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.mr.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.ms.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.9.ewk.1 | $168$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 168.192.9.bob.1 | $168$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 280.240.17.yq.1 | $280$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 280.288.17.dfk.1 | $280$ | $6$ | $6$ | $17$ | $?$ | not computed |