Invariants
Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $3136$ | ||
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: | $1$ | ||||||
$\Q$-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: | 56.48.1.152 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}1&4\\32&1\end{bmatrix}$, $\begin{bmatrix}30&43\\23&46\end{bmatrix}$, $\begin{bmatrix}49&30\\54&33\end{bmatrix}$, $\begin{bmatrix}55&8\\12&55\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 56.96.1-56.dn.1.1, 56.96.1-56.dn.1.2, 56.96.1-56.dn.1.3, 56.96.1-56.dn.1.4, 112.96.1-56.dn.1.1, 112.96.1-56.dn.1.2, 112.96.1-56.dn.1.3, 112.96.1-56.dn.1.4, 168.96.1-56.dn.1.1, 168.96.1-56.dn.1.2, 168.96.1-56.dn.1.3, 168.96.1-56.dn.1.4, 280.96.1-56.dn.1.1, 280.96.1-56.dn.1.2, 280.96.1-56.dn.1.3, 280.96.1-56.dn.1.4 |
Cyclic 56-isogeny field degree: | $16$ |
Cyclic 56-torsion field degree: | $384$ |
Full 56-torsion field degree: | $64512$ |
Jacobian
Conductor: | $2^{6}\cdot7^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 3136.2.a.m |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x y - z^{2} $ |
$=$ | $7 x^{2} + 7 y^{2} + 16 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 7 x^{2} y^{2} + z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{4}{7}w$ |
$\displaystyle Z$ | $=$ | $\displaystyle z$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{1}{7^2}\cdot\frac{(7z^{2}-4w^{2})^{3}(7z^{2}+4w^{2})^{3}}{w^{8}z^{4}}$ |
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 |
---|---|---|---|---|---|---|
8.24.0.t.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.0.u.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.0.dm.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.0.dn.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.1.n.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.24.1.cg.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.24.1.ch.1 | $56$ | $2$ | $2$ | $1$ | $1$ | 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.jj.1 | $56$ | $8$ | $8$ | $25$ | $9$ | $1^{20}\cdot2^{2}$ |
56.1008.73.zj.1 | $56$ | $21$ | $21$ | $73$ | $26$ | $1^{16}\cdot2^{26}\cdot4$ |
56.1344.97.yt.1 | $56$ | $28$ | $28$ | $97$ | $34$ | $1^{36}\cdot2^{28}\cdot4$ |
112.96.3.gt.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
112.96.3.gt.2 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
112.96.3.gu.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
112.96.3.gu.2 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.144.9.ctf.1 | $168$ | $3$ | $3$ | $9$ | $?$ | not computed |
168.192.9.bao.1 | $168$ | $4$ | $4$ | $9$ | $?$ | not computed |
280.240.17.ox.1 | $280$ | $5$ | $5$ | $17$ | $?$ | not computed |
280.288.17.bwj.1 | $280$ | $6$ | $6$ | $17$ | $?$ | not computed |