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: | $1 \le \gamma \le 2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8G0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.48.0.542 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}12&27\\21&18\end{bmatrix}$, $\begin{bmatrix}19&2\\36&45\end{bmatrix}$, $\begin{bmatrix}37&32\\40&1\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 56.24.0.bg.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 $ | $=$ | $ 14 x^{2} - 16 y^{2} + 7 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 |
---|---|---|---|---|---|
8.24.0-8.m.1.8 | $8$ | $2$ | $2$ | $0$ | $0$ |
56.24.0-8.m.1.1 | $56$ | $2$ | $2$ | $0$ | $0$ |
56.24.0-56.s.1.4 | $56$ | $2$ | $2$ | $0$ | $0$ |
56.24.0-56.s.1.8 | $56$ | $2$ | $2$ | $0$ | $0$ |
56.24.0-56.ba.1.4 | $56$ | $2$ | $2$ | $0$ | $0$ |
56.24.0-56.ba.1.16 | $56$ | $2$ | $2$ | $0$ | $0$ |
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.dw.1.10 | $56$ | $8$ | $8$ | $11$ |
56.1008.34-56.fa.1.7 | $56$ | $21$ | $21$ | $34$ |
56.1344.45-56.fe.1.15 | $56$ | $28$ | $28$ | $45$ |
168.144.4-168.is.1.31 | $168$ | $3$ | $3$ | $4$ |
168.192.3-168.lq.1.31 | $168$ | $4$ | $4$ | $3$ |
280.240.8-280.ds.1.16 | $280$ | $5$ | $5$ | $8$ |
280.288.7-280.hd.1.16 | $280$ | $6$ | $6$ | $7$ |
280.480.15-280.je.1.15 | $280$ | $10$ | $10$ | $15$ |