Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $448$ | ||
| Index: | $768$ | $\PSL_2$-index: | $768$ | ||||
| Genus: | $53 = 1 + \frac{ 768 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (of which $8$ are rational) | Cusp widths | $8^{12}\cdot56^{12}$ | Cusp orbits | $1^{8}\cdot2^{6}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $8 \le \gamma \le 16$ | ||||||
| $\overline{\Q}$-gonality: | $8 \le \gamma \le 16$ | ||||||
| Rational cusps: | $8$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.768.53.122 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}7&44\\8&7\end{bmatrix}$, $\begin{bmatrix}9&20\\28&39\end{bmatrix}$, $\begin{bmatrix}29&48\\28&17\end{bmatrix}$, $\begin{bmatrix}41&40\\32&33\end{bmatrix}$, $\begin{bmatrix}47&36\\48&41\end{bmatrix}$, $\begin{bmatrix}53&24\\40&25\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 56.1536.53-56.cy.2.1, 56.1536.53-56.cy.2.2, 56.1536.53-56.cy.2.3, 56.1536.53-56.cy.2.4, 56.1536.53-56.cy.2.5, 56.1536.53-56.cy.2.6, 56.1536.53-56.cy.2.7, 56.1536.53-56.cy.2.8, 56.1536.53-56.cy.2.9, 56.1536.53-56.cy.2.10, 56.1536.53-56.cy.2.11, 56.1536.53-56.cy.2.12, 56.1536.53-56.cy.2.13, 56.1536.53-56.cy.2.14, 56.1536.53-56.cy.2.15, 56.1536.53-56.cy.2.16, 56.1536.53-56.cy.2.17, 56.1536.53-56.cy.2.18, 56.1536.53-56.cy.2.19, 56.1536.53-56.cy.2.20, 56.1536.53-56.cy.2.21, 56.1536.53-56.cy.2.22, 56.1536.53-56.cy.2.23, 56.1536.53-56.cy.2.24 |
| Cyclic 56-isogeny field degree: | $2$ |
| Cyclic 56-torsion field degree: | $24$ |
| Full 56-torsion field degree: | $4032$ |
Jacobian
| Conductor: | $2^{220}\cdot7^{53}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{11}\cdot2^{5}\cdot4^{2}\cdot8^{3}$ |
| Newforms: | 14.2.a.a$^{4}$, 28.2.d.a, 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, 112.2.f.a, 112.2.f.b, 224.2.f.a, 448.2.e.a, 448.2.e.b |
Rational points
This modular curve has 8 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.384.23.i.1 | $56$ | $2$ | $2$ | $23$ | $1$ | $2^{3}\cdot8^{3}$ |
| 56.384.25.bl.2 | $56$ | $2$ | $2$ | $25$ | $1$ | $2^{2}\cdot4^{2}\cdot8^{2}$ |
| 56.384.27.c.2 | $56$ | $2$ | $2$ | $27$ | $1$ | $2^{5}\cdot4^{2}\cdot8$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.1536.105.cj.1 | $56$ | $2$ | $2$ | $105$ | $3$ | $1^{10}\cdot2^{13}\cdot4^{4}$ |
| 56.1536.105.ck.1 | $56$ | $2$ | $2$ | $105$ | $8$ | $1^{10}\cdot2^{13}\cdot4^{4}$ |
| 56.1536.105.cn.2 | $56$ | $2$ | $2$ | $105$ | $7$ | $1^{10}\cdot2^{13}\cdot4^{4}$ |
| 56.1536.105.co.2 | $56$ | $2$ | $2$ | $105$ | $10$ | $1^{10}\cdot2^{13}\cdot4^{4}$ |
| 56.1536.105.gm.1 | $56$ | $2$ | $2$ | $105$ | $6$ | $1^{10}\cdot2^{9}\cdot4^{4}\cdot8$ |
| 56.1536.105.gm.3 | $56$ | $2$ | $2$ | $105$ | $6$ | $1^{10}\cdot2^{9}\cdot4^{4}\cdot8$ |
| 56.1536.105.gn.2 | $56$ | $2$ | $2$ | $105$ | $8$ | $1^{10}\cdot2^{9}\cdot4^{4}\cdot8$ |
| 56.1536.105.gn.4 | $56$ | $2$ | $2$ | $105$ | $8$ | $1^{10}\cdot2^{9}\cdot4^{4}\cdot8$ |
| 56.1536.105.gq.6 | $56$ | $2$ | $2$ | $105$ | $9$ | $1^{10}\cdot2^{9}\cdot4^{4}\cdot8$ |
| 56.1536.105.gq.8 | $56$ | $2$ | $2$ | $105$ | $9$ | $1^{10}\cdot2^{9}\cdot4^{4}\cdot8$ |
| 56.1536.105.gr.5 | $56$ | $2$ | $2$ | $105$ | $5$ | $1^{10}\cdot2^{9}\cdot4^{4}\cdot8$ |
| 56.1536.105.gr.7 | $56$ | $2$ | $2$ | $105$ | $5$ | $1^{10}\cdot2^{9}\cdot4^{4}\cdot8$ |
| 56.1536.105.ir.2 | $56$ | $2$ | $2$ | $105$ | $3$ | $1^{10}\cdot2^{13}\cdot4^{4}$ |
| 56.1536.105.is.2 | $56$ | $2$ | $2$ | $105$ | $8$ | $1^{10}\cdot2^{13}\cdot4^{4}$ |
| 56.1536.105.iv.1 | $56$ | $2$ | $2$ | $105$ | $7$ | $1^{10}\cdot2^{13}\cdot4^{4}$ |
| 56.1536.105.iw.1 | $56$ | $2$ | $2$ | $105$ | $10$ | $1^{10}\cdot2^{13}\cdot4^{4}$ |
| 56.2304.157.fe.1 | $56$ | $3$ | $3$ | $157$ | $1$ | $2^{12}\cdot4^{2}\cdot8\cdot12^{4}\cdot16$ |
| 56.2304.157.fg.2 | $56$ | $3$ | $3$ | $157$ | $1$ | $2^{12}\cdot4^{2}\cdot8\cdot12^{4}\cdot16$ |
| 56.2304.157.hn.2 | $56$ | $3$ | $3$ | $157$ | $11$ | $1^{20}\cdot2^{2}\cdot4^{2}\cdot6^{4}\cdot8\cdot12^{2}\cdot16$ |
| 56.5376.401.fd.1 | $56$ | $7$ | $7$ | $401$ | $20$ | $1^{48}\cdot2^{26}\cdot4^{10}\cdot6^{4}\cdot8^{9}\cdot12^{4}\cdot16^{2}\cdot32$ |