Invariants
Level: | $56$ | $\SL_2$-level: | $8$ | ||||
Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $2^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 4E0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.24.0.274 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}14&51\\41&16\end{bmatrix}$, $\begin{bmatrix}43&30\\22&49\end{bmatrix}$, $\begin{bmatrix}44&37\\45&54\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 56.12.0.m.1 for the level structure with $-I$) |
Cyclic 56-isogeny field degree: | $32$ |
Cyclic 56-torsion field degree: | $768$ |
Full 56-torsion field degree: | $129024$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ 72 x^{2} - 7 y^{2} - 448 z^{2} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
8.12.0-4.b.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ |
56.12.0-4.b.1.3 | $56$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
56.192.5-56.y.1.12 | $56$ | $8$ | $8$ | $5$ |
56.504.16-56.bw.1.8 | $56$ | $21$ | $21$ | $16$ |
56.672.21-56.bw.1.7 | $56$ | $28$ | $28$ | $21$ |
168.72.2-168.bw.1.16 | $168$ | $3$ | $3$ | $2$ |
168.96.1-168.yu.1.24 | $168$ | $4$ | $4$ | $1$ |
280.120.4-280.y.1.8 | $280$ | $5$ | $5$ | $4$ |
280.144.3-280.bk.1.16 | $280$ | $6$ | $6$ | $3$ |
280.240.7-280.bw.1.16 | $280$ | $10$ | $10$ | $7$ |