Invariants
Level: | $56$ | $\SL_2$-level: | $8$ | ||||
Index: | $48$ | $\PSL_2$-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: | 56.48.0.512 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}7&10\\16&1\end{bmatrix}$, $\begin{bmatrix}30&5\\51&4\end{bmatrix}$, $\begin{bmatrix}55&46\\46&39\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 56.24.0.bm.1 for the level structure with $-I$) |
Cyclic 56-isogeny field degree: | $16$ |
Cyclic 56-torsion field degree: | $192$ |
Full 56-torsion field degree: | $64512$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ 7 x^{2} + 16 y^{2} + 7 z^{2} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
8.24.0-8.o.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
28.24.0-28.g.1.2 | $28$ | $2$ | $2$ | $0$ | $0$ |
56.24.0-28.g.1.5 | $56$ | $2$ | $2$ | $0$ | $0$ |
56.24.0-8.o.1.6 | $56$ | $2$ | $2$ | $0$ | $0$ |
56.24.0-56.z.1.2 | $56$ | $2$ | $2$ | $0$ | $0$ |
56.24.0-56.z.1.13 | $56$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
56.384.11-56.eo.1.1 | $56$ | $8$ | $8$ | $11$ |
56.1008.34-56.fo.1.2 | $56$ | $21$ | $21$ | $34$ |
56.1344.45-56.fs.1.2 | $56$ | $28$ | $28$ | $45$ |
112.96.1-112.y.1.8 | $112$ | $2$ | $2$ | $1$ |
112.96.1-112.ba.1.6 | $112$ | $2$ | $2$ | $1$ |
112.96.1-112.cm.1.5 | $112$ | $2$ | $2$ | $1$ |
112.96.1-112.co.1.6 | $112$ | $2$ | $2$ | $1$ |
168.144.4-168.ka.1.1 | $168$ | $3$ | $3$ | $4$ |
168.192.3-168.mi.1.1 | $168$ | $4$ | $4$ | $3$ |
280.240.8-280.ek.1.1 | $280$ | $5$ | $5$ | $8$ |
280.288.7-280.ih.1.1 | $280$ | $6$ | $6$ | $7$ |
280.480.15-280.ku.1.1 | $280$ | $10$ | $10$ | $15$ |