Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $1568$ | ||
| Index: | $1344$ | $\PSL_2$-index: | $672$ | ||||
| Genus: | $49 = 1 + \frac{ 672 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $2$ are rational) | Cusp widths | $28^{8}\cdot56^{8}$ | Cusp orbits | $1^{2}\cdot2\cdot3^{2}\cdot6$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $17$ | ||||||
| $\Q$-gonality: | $12 \le \gamma \le 24$ | ||||||
| $\overline{\Q}$-gonality: | $12 \le \gamma \le 24$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.1344.49.528 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}4&49\\21&4\end{bmatrix}$, $\begin{bmatrix}16&7\\7&44\end{bmatrix}$, $\begin{bmatrix}29&20\\28&33\end{bmatrix}$, $\begin{bmatrix}43&46\\42&55\end{bmatrix}$, $\begin{bmatrix}46&11\\19&10\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.672.49.by.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $4$ |
| Cyclic 56-torsion field degree: | $96$ |
| Full 56-torsion field degree: | $2304$ |
Jacobian
| Conductor: | $2^{200}\cdot7^{93}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{13}\cdot2^{16}\cdot4$ |
| Newforms: | 14.2.a.a$^{2}$, 98.2.a.b$^{2}$, 112.2.a.a, 112.2.a.b, 112.2.a.c, 196.2.a.b, 196.2.a.c, 784.2.a.a, 784.2.a.d, 784.2.a.h, 784.2.a.k, 784.2.a.l, 784.2.a.m, 1568.2.a.b, 1568.2.a.e$^{2}$, 1568.2.a.h, 1568.2.a.j, 1568.2.a.k, 1568.2.a.n, 1568.2.a.o, 1568.2.a.p, 1568.2.a.q, 1568.2.a.r, 1568.2.a.t, 1568.2.a.u, 1568.2.a.v, 1568.2.a.x |
Rational points
This modular curve has 2 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 |
|---|---|---|---|---|---|---|
| 56.48.1-56.m.1.6 | $56$ | $28$ | $28$ | $1$ | $1$ | $1^{12}\cdot2^{16}\cdot4$ |
| 56.672.21-28.p.1.1 | $56$ | $2$ | $2$ | $21$ | $5$ | $1^{4}\cdot2^{10}\cdot4$ |
| 56.672.21-28.p.1.21 | $56$ | $2$ | $2$ | $21$ | $5$ | $1^{4}\cdot2^{10}\cdot4$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.2688.97-56.uh.1.1 | $56$ | $2$ | $2$ | $97$ | $31$ | $1^{24}\cdot2^{12}$ |
| 56.2688.97-56.uo.1.3 | $56$ | $2$ | $2$ | $97$ | $26$ | $1^{24}\cdot2^{12}$ |
| 56.2688.97-56.yl.1.7 | $56$ | $2$ | $2$ | $97$ | $40$ | $1^{24}\cdot2^{12}$ |
| 56.2688.97-56.yq.1.5 | $56$ | $2$ | $2$ | $97$ | $29$ | $1^{24}\cdot2^{12}$ |
| 56.2688.97-56.zq.1.12 | $56$ | $2$ | $2$ | $97$ | $32$ | $1^{30}\cdot2^{9}$ |
| 56.2688.97-56.zw.1.6 | $56$ | $2$ | $2$ | $97$ | $38$ | $1^{30}\cdot2^{9}$ |
| 56.2688.97-56.bal.1.6 | $56$ | $2$ | $2$ | $97$ | $30$ | $1^{40}\cdot2^{4}$ |
| 56.2688.97-56.bat.1.6 | $56$ | $2$ | $2$ | $97$ | $32$ | $1^{40}\cdot2^{4}$ |
| 56.2688.97-56.bbo.1.12 | $56$ | $2$ | $2$ | $97$ | $32$ | $1^{30}\cdot2^{9}$ |
| 56.2688.97-56.bbu.1.7 | $56$ | $2$ | $2$ | $97$ | $40$ | $1^{30}\cdot2^{9}$ |
| 56.2688.97-56.bcj.1.8 | $56$ | $2$ | $2$ | $97$ | $36$ | $1^{40}\cdot2^{4}$ |
| 56.2688.97-56.bcr.1.7 | $56$ | $2$ | $2$ | $97$ | $34$ | $1^{40}\cdot2^{4}$ |
| 56.2688.97-56.bdp.1.7 | $56$ | $2$ | $2$ | $97$ | $28$ | $1^{40}\cdot2^{4}$ |
| 56.2688.97-56.bdx.1.8 | $56$ | $2$ | $2$ | $97$ | $42$ | $1^{40}\cdot2^{4}$ |
| 56.2688.97-56.bes.1.7 | $56$ | $2$ | $2$ | $97$ | $39$ | $1^{30}\cdot2^{9}$ |
| 56.2688.97-56.bew.1.7 | $56$ | $2$ | $2$ | $97$ | $34$ | $1^{30}\cdot2^{9}$ |
| 56.2688.97-56.bfl.1.5 | $56$ | $2$ | $2$ | $97$ | $30$ | $1^{40}\cdot2^{4}$ |
| 56.2688.97-56.bft.1.5 | $56$ | $2$ | $2$ | $97$ | $48$ | $1^{40}\cdot2^{4}$ |
| 56.2688.97-56.bgo.1.6 | $56$ | $2$ | $2$ | $97$ | $41$ | $1^{30}\cdot2^{9}$ |
| 56.2688.97-56.bgs.1.6 | $56$ | $2$ | $2$ | $97$ | $36$ | $1^{30}\cdot2^{9}$ |
| 56.2688.97-56.bhc.1.7 | $56$ | $2$ | $2$ | $97$ | $31$ | $1^{24}\cdot2^{12}$ |
| 56.2688.97-56.bhn.1.5 | $56$ | $2$ | $2$ | $97$ | $40$ | $1^{24}\cdot2^{12}$ |
| 56.2688.97-56.bik.1.3 | $56$ | $2$ | $2$ | $97$ | $34$ | $1^{24}\cdot2^{12}$ |
| 56.2688.97-56.bit.1.2 | $56$ | $2$ | $2$ | $97$ | $49$ | $1^{24}\cdot2^{12}$ |
| 56.4032.145-56.czy.1.10 | $56$ | $3$ | $3$ | $145$ | $50$ | $1^{38}\cdot2^{27}\cdot4$ |