Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $448$ | ||
| Index: | $768$ | $\PSL_2$-index: | $768$ | ||||
| Genus: | $45 = 1 + \frac{ 768 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 40 }{2}$ | ||||||
| Cusps: | $40$ (none of which are rational) | Cusp widths | $4^{16}\cdot8^{4}\cdot28^{16}\cdot56^{4}$ | Cusp orbits | $2^{8}\cdot4^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $3$ | ||||||
| $\Q$-gonality: | $9 \le \gamma \le 16$ | ||||||
| $\overline{\Q}$-gonality: | $9 \le \gamma \le 16$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.768.45.338 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}17&24\\8&35\end{bmatrix}$, $\begin{bmatrix}17&44\\20&49\end{bmatrix}$, $\begin{bmatrix}23&8\\28&5\end{bmatrix}$, $\begin{bmatrix}23&40\\12&41\end{bmatrix}$, $\begin{bmatrix}33&32\\4&11\end{bmatrix}$, $\begin{bmatrix}47&8\\32&9\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 56.1536.45-56.bf.1.1, 56.1536.45-56.bf.1.2, 56.1536.45-56.bf.1.3, 56.1536.45-56.bf.1.4, 56.1536.45-56.bf.1.5, 56.1536.45-56.bf.1.6, 56.1536.45-56.bf.1.7, 56.1536.45-56.bf.1.8, 56.1536.45-56.bf.1.9, 56.1536.45-56.bf.1.10, 56.1536.45-56.bf.1.11, 56.1536.45-56.bf.1.12, 56.1536.45-56.bf.1.13, 56.1536.45-56.bf.1.14, 56.1536.45-56.bf.1.15, 56.1536.45-56.bf.1.16, 56.1536.45-56.bf.1.17, 56.1536.45-56.bf.1.18, 56.1536.45-56.bf.1.19, 56.1536.45-56.bf.1.20 |
| Cyclic 56-isogeny field degree: | $2$ |
| Cyclic 56-torsion field degree: | $24$ |
| Full 56-torsion field degree: | $4032$ |
Jacobian
| Conductor: | $2^{184}\cdot7^{45}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{11}\cdot2^{9}\cdot4^{4}$ |
| Newforms: | 14.2.a.a$^{4}$, 56.2.a.a$^{2}$, 56.2.a.b$^{2}$, 56.2.b.a$^{2}$, 56.2.b.b$^{2}$, 112.2.a.a, 112.2.a.b, 112.2.a.c, 224.2.e.a$^{2}$, 224.2.e.b$^{2}$, 448.2.f.a, 448.2.f.b$^{3}$, 448.2.f.c |
Rational points
This modular curve has no $\Q_p$ points for $p=53,107,211,317$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.384.21.d.2 | $56$ | $2$ | $2$ | $21$ | $3$ | $2^{4}\cdot4^{4}$ |
| 56.384.21.bo.1 | $56$ | $2$ | $2$ | $21$ | $0$ | $1^{6}\cdot2^{5}\cdot4^{2}$ |
| 56.384.21.bo.2 | $56$ | $2$ | $2$ | $21$ | $0$ | $1^{6}\cdot2^{5}\cdot4^{2}$ |
| 56.384.23.i.1 | $56$ | $2$ | $2$ | $23$ | $1$ | $2^{7}\cdot4^{2}$ |
| 56.384.23.q.2 | $56$ | $2$ | $2$ | $23$ | $1$ | $2^{7}\cdot4^{2}$ |
| 56.384.23.ch.1 | $56$ | $2$ | $2$ | $23$ | $2$ | $1^{6}\cdot2^{4}\cdot4^{2}$ |
| 56.384.23.ch.2 | $56$ | $2$ | $2$ | $23$ | $2$ | $1^{6}\cdot2^{4}\cdot4^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.1536.97.fy.1 | $56$ | $2$ | $2$ | $97$ | $8$ | $1^{10}\cdot2^{9}\cdot4^{4}\cdot8$ |
| 56.1536.97.fy.4 | $56$ | $2$ | $2$ | $97$ | $8$ | $1^{10}\cdot2^{9}\cdot4^{4}\cdot8$ |
| 56.1536.97.fz.2 | $56$ | $2$ | $2$ | $97$ | $10$ | $1^{10}\cdot2^{9}\cdot4^{4}\cdot8$ |
| 56.1536.97.fz.3 | $56$ | $2$ | $2$ | $97$ | $10$ | $1^{10}\cdot2^{9}\cdot4^{4}\cdot8$ |
| 56.1536.97.ga.2 | $56$ | $2$ | $2$ | $97$ | $11$ | $1^{10}\cdot2^{9}\cdot4^{4}\cdot8$ |
| 56.1536.97.ga.3 | $56$ | $2$ | $2$ | $97$ | $11$ | $1^{10}\cdot2^{9}\cdot4^{4}\cdot8$ |
| 56.1536.97.gb.1 | $56$ | $2$ | $2$ | $97$ | $7$ | $1^{10}\cdot2^{9}\cdot4^{4}\cdot8$ |
| 56.1536.97.gb.4 | $56$ | $2$ | $2$ | $97$ | $7$ | $1^{10}\cdot2^{9}\cdot4^{4}\cdot8$ |
| 56.1536.105.is.2 | $56$ | $2$ | $2$ | $105$ | $8$ | $1^{10}\cdot2^{9}\cdot4^{2}\cdot8^{3}$ |
| 56.1536.105.it.2 | $56$ | $2$ | $2$ | $105$ | $7$ | $1^{10}\cdot2^{9}\cdot4^{2}\cdot8^{3}$ |
| 56.1536.105.iw.2 | $56$ | $2$ | $2$ | $105$ | $10$ | $1^{10}\cdot2^{9}\cdot4^{2}\cdot8^{3}$ |
| 56.1536.105.ix.2 | $56$ | $2$ | $2$ | $105$ | $11$ | $1^{10}\cdot2^{9}\cdot4^{2}\cdot8^{3}$ |
| 56.2304.133.g.1 | $56$ | $3$ | $3$ | $133$ | $11$ | $1^{20}\cdot2^{2}\cdot4^{4}\cdot6^{4}\cdot12^{2}$ |
| 56.2304.133.s.1 | $56$ | $3$ | $3$ | $133$ | $3$ | $2^{12}\cdot4^{4}\cdot12^{4}$ |
| 56.2304.133.be.2 | $56$ | $3$ | $3$ | $133$ | $5$ | $2^{12}\cdot4^{4}\cdot12^{4}$ |
| 56.5376.369.cd.1 | $56$ | $7$ | $7$ | $369$ | $22$ | $1^{48}\cdot2^{30}\cdot4^{20}\cdot6^{4}\cdot8^{8}\cdot12^{4}$ |