Invariants
Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $3136$ | ||
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: | $1$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8C1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.24.1.50 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}12&47\\33&8\end{bmatrix}$, $\begin{bmatrix}14&43\\25&18\end{bmatrix}$, $\begin{bmatrix}47&16\\48&41\end{bmatrix}$, $\begin{bmatrix}53&44\\44&47\end{bmatrix}$ |
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: | $129024$ |
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 $ | $=$ | $ 2 x z - y z + 2 y w $ |
$=$ | $8 x^{2} + 7 y^{2} + 4 z^{2} - 2 z w + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 4 x^{4} - 2 x^{3} z + 9 x^{2} y^{2} + 2 x^{2} z^{2} - 8 x y^{2} z + 8 y^{2} z^{2} $ |
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:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle y$ |
$\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^6\cdot3^3\,\frac{52009776xy^{3}w^{2}-47624864xyw^{4}+20253807y^{6}-32506110y^{4}w^{2}-9352924y^{2}w^{4}+2426112z^{6}+2765664z^{5}w+474192z^{4}w^{2}+9062728z^{3}w^{3}-7169408z^{2}w^{4}+13323392zw^{5}-472392w^{6}}{4000752xy^{3}w^{2}+2820832xyw^{4}+20253807y^{6}-2500470y^{4}w^{2}+1793204y^{2}w^{4}+1446336z^{6}+4507488z^{5}w-5598000z^{4}w^{2}+4947400z^{3}w^{3}-3037280z^{2}w^{4}+711968zw^{5}-472392w^{6}}$ |
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 |
---|---|---|---|---|---|---|
8.12.0.q.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
28.12.0.n.1 | $28$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.12.1.c.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 |
---|---|---|---|---|---|---|
56.48.1.bp.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.48.1.cc.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.48.1.cv.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.48.1.da.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.48.1.hl.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.48.1.hp.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.48.1.ia.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.48.1.ie.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.192.13.dl.1 | $56$ | $8$ | $8$ | $13$ | $7$ | $1^{12}$ |
56.504.37.jx.1 | $56$ | $21$ | $21$ | $37$ | $11$ | $1^{8}\cdot2^{12}\cdot4$ |
56.672.49.jx.1 | $56$ | $28$ | $28$ | $49$ | $17$ | $1^{20}\cdot2^{12}\cdot4$ |
168.48.1.baz.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.bbh.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.bbp.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.bbx.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.cfd.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.cfl.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.cfs.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.cga.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.72.5.yz.1 | $168$ | $3$ | $3$ | $5$ | $?$ | not computed |
168.96.5.od.1 | $168$ | $4$ | $4$ | $5$ | $?$ | not computed |
280.48.1.xz.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.yh.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.yp.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.yx.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.bzj.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.bzr.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.bzy.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.cag.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.120.9.hl.1 | $280$ | $5$ | $5$ | $9$ | $?$ | not computed |
280.144.9.vp.1 | $280$ | $6$ | $6$ | $9$ | $?$ | not computed |
280.240.17.dkd.1 | $280$ | $10$ | $10$ | $17$ | $?$ | not computed |