Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $112$ | ||
| Index: | $768$ | $\PSL_2$-index: | $384$ | ||||
| Genus: | $23 = 1 + \frac{ 384 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 20 }{2}$ | ||||||
| Cusps: | $20$ (of which $8$ are rational) | Cusp widths | $4^{8}\cdot8^{2}\cdot28^{8}\cdot56^{2}$ | Cusp orbits | $1^{8}\cdot2^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $5 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $5 \le \gamma \le 8$ | ||||||
| Rational cusps: | $8$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 56P23 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.768.23.6472 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}29&36\\44&49\end{bmatrix}$, $\begin{bmatrix}31&24\\44&47\end{bmatrix}$, $\begin{bmatrix}39&24\\0&33\end{bmatrix}$, $\begin{bmatrix}39&52\\52&17\end{bmatrix}$, $\begin{bmatrix}41&32\\32&7\end{bmatrix}$, $\begin{bmatrix}55&16\\36&55\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.384.23.i.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $2$ |
| Cyclic 56-torsion field degree: | $24$ |
| Full 56-torsion field degree: | $4032$ |
Jacobian
| Conductor: | $2^{64}\cdot7^{23}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{11}\cdot2^{2}\cdot4^{2}$ |
| 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 |
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 |
|---|---|---|---|---|---|---|
| 56.384.11-28.b.1.27 | $56$ | $2$ | $2$ | $11$ | $1$ | $2^{2}\cdot4^{2}$ |
| 56.384.11-28.b.1.28 | $56$ | $2$ | $2$ | $11$ | $1$ | $2^{2}\cdot4^{2}$ |
| 56.384.11-56.s.1.5 | $56$ | $2$ | $2$ | $11$ | $0$ | $1^{6}\cdot2\cdot4$ |
| 56.384.11-56.s.1.60 | $56$ | $2$ | $2$ | $11$ | $0$ | $1^{6}\cdot2\cdot4$ |
| 56.384.11-56.s.2.10 | $56$ | $2$ | $2$ | $11$ | $0$ | $1^{6}\cdot2\cdot4$ |
| 56.384.11-56.s.2.55 | $56$ | $2$ | $2$ | $11$ | $0$ | $1^{6}\cdot2\cdot4$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.1536.45-56.g.1.1 | $56$ | $2$ | $2$ | $45$ | $1$ | $2^{7}\cdot4^{2}$ |
| 56.1536.45-56.g.2.1 | $56$ | $2$ | $2$ | $45$ | $1$ | $2^{7}\cdot4^{2}$ |
| 56.1536.45-56.h.1.2 | $56$ | $2$ | $2$ | $45$ | $3$ | $2^{7}\cdot4^{2}$ |
| 56.1536.45-56.h.2.2 | $56$ | $2$ | $2$ | $45$ | $3$ | $2^{7}\cdot4^{2}$ |
| 56.1536.45-56.be.1.6 | $56$ | $2$ | $2$ | $45$ | $1$ | $2^{7}\cdot4^{2}$ |
| 56.1536.45-56.be.2.6 | $56$ | $2$ | $2$ | $45$ | $1$ | $2^{7}\cdot4^{2}$ |
| 56.1536.45-56.bf.1.5 | $56$ | $2$ | $2$ | $45$ | $3$ | $2^{7}\cdot4^{2}$ |
| 56.1536.45-56.bf.2.5 | $56$ | $2$ | $2$ | $45$ | $3$ | $2^{7}\cdot4^{2}$ |
| 56.1536.49-56.fr.1.11 | $56$ | $2$ | $2$ | $49$ | $6$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.1536.49-56.fr.2.14 | $56$ | $2$ | $2$ | $49$ | $6$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.1536.49-56.fs.1.12 | $56$ | $2$ | $2$ | $49$ | $8$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.1536.49-56.fs.2.6 | $56$ | $2$ | $2$ | $49$ | $8$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.1536.49-56.ft.1.7 | $56$ | $2$ | $2$ | $49$ | $1$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.1536.49-56.ft.2.9 | $56$ | $2$ | $2$ | $49$ | $1$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.1536.49-56.ft.3.13 | $56$ | $2$ | $2$ | $49$ | $1$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.1536.49-56.ft.4.14 | $56$ | $2$ | $2$ | $49$ | $1$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.1536.49-56.fu.1.5 | $56$ | $2$ | $2$ | $49$ | $3$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.1536.49-56.fu.2.11 | $56$ | $2$ | $2$ | $49$ | $3$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.1536.49-56.fu.3.14 | $56$ | $2$ | $2$ | $49$ | $3$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.1536.49-56.fu.4.10 | $56$ | $2$ | $2$ | $49$ | $3$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.1536.49-56.fv.1.12 | $56$ | $2$ | $2$ | $49$ | $7$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.1536.49-56.fv.2.4 | $56$ | $2$ | $2$ | $49$ | $7$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.1536.49-56.fw.1.10 | $56$ | $2$ | $2$ | $49$ | $3$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.1536.49-56.fw.2.12 | $56$ | $2$ | $2$ | $49$ | $3$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.1536.53-56.cu.1.12 | $56$ | $2$ | $2$ | $53$ | $6$ | $1^{10}\cdot2^{6}\cdot4^{2}$ |
| 56.1536.53-56.cv.1.12 | $56$ | $2$ | $2$ | $53$ | $3$ | $1^{10}\cdot2^{6}\cdot4^{2}$ |
| 56.1536.53-56.cw.1.12 | $56$ | $2$ | $2$ | $53$ | $8$ | $1^{10}\cdot2^{6}\cdot4^{2}$ |
| 56.1536.53-56.cx.1.12 | $56$ | $2$ | $2$ | $53$ | $7$ | $1^{10}\cdot2^{6}\cdot4^{2}$ |
| 56.1536.53-56.cy.1.18 | $56$ | $2$ | $2$ | $53$ | $1$ | $2^{3}\cdot8^{3}$ |
| 56.1536.53-56.cy.2.18 | $56$ | $2$ | $2$ | $53$ | $1$ | $2^{3}\cdot8^{3}$ |
| 56.1536.53-56.cz.1.21 | $56$ | $2$ | $2$ | $53$ | $3$ | $2^{3}\cdot8^{3}$ |
| 56.1536.53-56.cz.2.21 | $56$ | $2$ | $2$ | $53$ | $3$ | $2^{3}\cdot8^{3}$ |
| 56.2304.67-56.n.1.11 | $56$ | $3$ | $3$ | $67$ | $1$ | $2^{10}\cdot12^{2}$ |
| 56.2304.67-56.n.2.18 | $56$ | $3$ | $3$ | $67$ | $1$ | $2^{10}\cdot12^{2}$ |
| 56.2304.67-56.bc.1.10 | $56$ | $3$ | $3$ | $67$ | $9$ | $1^{20}\cdot6^{4}$ |
| 56.5376.185-56.m.1.22 | $56$ | $7$ | $7$ | $185$ | $18$ | $1^{48}\cdot2^{21}\cdot4^{6}\cdot6^{4}\cdot12^{2}$ |