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: | $2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 4E0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.24.0.267 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}10&5\\23&40\end{bmatrix}$, $\begin{bmatrix}22&29\\27&16\end{bmatrix}$, $\begin{bmatrix}33&54\\6&39\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 28.12.0.e.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 $ | $=$ | $ 9 x^{2} + 3 x y + 2 y^{2} + 112 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 |
---|---|---|---|---|---|
8.12.0-4.b.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
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 |
---|---|---|---|---|
56.192.5-28.i.1.7 | $56$ | $8$ | $8$ | $5$ |
56.504.16-28.q.1.4 | $56$ | $21$ | $21$ | $16$ |
56.672.21-28.q.1.2 | $56$ | $28$ | $28$ | $21$ |
168.72.2-84.q.1.4 | $168$ | $3$ | $3$ | $2$ |
168.96.1-84.i.1.11 | $168$ | $4$ | $4$ | $1$ |
280.120.4-140.i.1.5 | $280$ | $5$ | $5$ | $4$ |
280.144.3-140.m.1.4 | $280$ | $6$ | $6$ | $3$ |
280.240.7-140.q.1.2 | $280$ | $10$ | $10$ | $7$ |