Invariants
Level: | $56$ | $\SL_2$-level: | $8$ | ||||
Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
Cusps: | $6$ (none of which are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | Cusp orbits | $2^{3}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8G0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.24.0.69 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}3&6\\46&35\end{bmatrix}$, $\begin{bmatrix}24&37\\21&24\end{bmatrix}$, $\begin{bmatrix}25&50\\26&37\end{bmatrix}$, $\begin{bmatrix}46&31\\15&18\end{bmatrix}$, $\begin{bmatrix}47&4\\32&7\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 56.48.0-56.u.1.1, 56.48.0-56.u.1.2, 56.48.0-56.u.1.3, 56.48.0-56.u.1.4, 56.48.0-56.u.1.5, 56.48.0-56.u.1.6, 56.48.0-56.u.1.7, 56.48.0-56.u.1.8, 168.48.0-56.u.1.1, 168.48.0-56.u.1.2, 168.48.0-56.u.1.3, 168.48.0-56.u.1.4, 168.48.0-56.u.1.5, 168.48.0-56.u.1.6, 168.48.0-56.u.1.7, 168.48.0-56.u.1.8, 280.48.0-56.u.1.1, 280.48.0-56.u.1.2, 280.48.0-56.u.1.3, 280.48.0-56.u.1.4, 280.48.0-56.u.1.5, 280.48.0-56.u.1.6, 280.48.0-56.u.1.7, 280.48.0-56.u.1.8 |
Cyclic 56-isogeny field degree: | $16$ |
Cyclic 56-torsion field degree: | $384$ |
Full 56-torsion field degree: | $129024$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ 2 x^{2} + 3 x y + 2 y^{2} + 28 z^{2} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
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 |
---|---|---|---|---|---|
4.12.0.d.1 | $4$ | $2$ | $2$ | $0$ | $0$ |
56.12.0.z.1 | $56$ | $2$ | $2$ | $0$ | $0$ |
56.12.0.bp.1 | $56$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
56.48.1.dk.1 | $56$ | $2$ | $2$ | $1$ |
56.48.1.dl.1 | $56$ | $2$ | $2$ | $1$ |
56.48.1.dm.1 | $56$ | $2$ | $2$ | $1$ |
56.48.1.dn.1 | $56$ | $2$ | $2$ | $1$ |
56.192.11.ct.1 | $56$ | $8$ | $8$ | $11$ |
56.504.34.dl.1 | $56$ | $21$ | $21$ | $34$ |
56.672.45.dn.1 | $56$ | $28$ | $28$ | $45$ |
168.48.1.km.1 | $168$ | $2$ | $2$ | $1$ |
168.48.1.kn.1 | $168$ | $2$ | $2$ | $1$ |
168.48.1.ko.1 | $168$ | $2$ | $2$ | $1$ |
168.48.1.kp.1 | $168$ | $2$ | $2$ | $1$ |
168.72.4.ft.1 | $168$ | $3$ | $3$ | $4$ |
168.96.3.hh.1 | $168$ | $4$ | $4$ | $3$ |
280.48.1.jo.1 | $280$ | $2$ | $2$ | $1$ |
280.48.1.jp.1 | $280$ | $2$ | $2$ | $1$ |
280.48.1.jq.1 | $280$ | $2$ | $2$ | $1$ |
280.48.1.jr.1 | $280$ | $2$ | $2$ | $1$ |
280.120.8.cb.1 | $280$ | $5$ | $5$ | $8$ |
280.144.7.ed.1 | $280$ | $6$ | $6$ | $7$ |
280.240.15.fv.1 | $280$ | $10$ | $10$ | $15$ |