Invariants
Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
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: | 56.24.1.69 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}3&26\\34&23\end{bmatrix}$, $\begin{bmatrix}39&38\\50&31\end{bmatrix}$, $\begin{bmatrix}44&1\\19&12\end{bmatrix}$, $\begin{bmatrix}47&38\\0&45\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 56.48.1-56.s.1.1, 56.48.1-56.s.1.2, 56.48.1-56.s.1.3, 56.48.1-56.s.1.4, 112.48.1-56.s.1.1, 112.48.1-56.s.1.2, 112.48.1-56.s.1.3, 112.48.1-56.s.1.4, 168.48.1-56.s.1.1, 168.48.1-56.s.1.2, 168.48.1-56.s.1.3, 168.48.1-56.s.1.4, 280.48.1-56.s.1.1, 280.48.1-56.s.1.2, 280.48.1-56.s.1.3, 280.48.1-56.s.1.4 |
Cyclic 56-isogeny field degree: | $32$ |
Cyclic 56-torsion field degree: | $768$ |
Full 56-torsion field degree: | $129024$ |
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} $ |
$=$ | $28 x^{2} + y w - 2 z w$ |
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 no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{14}w$ |
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.
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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.b.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
28.12.0.i.1 | $28$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.12.0.bu.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 |
---|---|---|---|---|---|---|
56.192.13.bo.1 | $56$ | $8$ | $8$ | $13$ | $3$ | $1^{8}\cdot2^{2}$ |
56.504.37.co.1 | $56$ | $21$ | $21$ | $37$ | $11$ | $1^{4}\cdot2^{14}\cdot4$ |
56.672.49.co.1 | $56$ | $28$ | $28$ | $49$ | $14$ | $1^{12}\cdot2^{16}\cdot4$ |
112.48.3.cn.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
112.48.3.co.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
112.48.3.cv.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
112.48.3.cw.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.5.co.1 | $168$ | $3$ | $3$ | $5$ | $?$ | not computed |
168.96.5.cm.1 | $168$ | $4$ | $4$ | $5$ | $?$ | not computed |
280.120.9.bi.1 | $280$ | $5$ | $5$ | $9$ | $?$ | not computed |
280.144.9.cg.1 | $280$ | $6$ | $6$ | $9$ | $?$ | not computed |
280.240.17.bhs.1 | $280$ | $10$ | $10$ | $17$ | $?$ | not computed |