Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
| Index: | $1008$ | $\PSL_2$-index: | $1008$ | ||||
| Genus: | $73 = 1 + \frac{ 1008 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (none of which are rational) | Cusp widths | $28^{12}\cdot56^{12}$ | Cusp orbits | $12^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $23$ | ||||||
| $\Q$-gonality: | $17 \le \gamma \le 32$ | ||||||
| $\overline{\Q}$-gonality: | $17 \le \gamma \le 32$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.1008.73.952 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}15&22\\36&11\end{bmatrix}$, $\begin{bmatrix}23&14\\20&19\end{bmatrix}$, $\begin{bmatrix}27&39\\0&1\end{bmatrix}$, $\begin{bmatrix}41&36\\22&51\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 56-isogeny field degree: | $32$ |
| Cyclic 56-torsion field degree: | $768$ |
| Full 56-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{360}\cdot7^{146}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{17}\cdot2^{26}\cdot4$ |
| Newforms: | 98.2.a.b$^{2}$, 196.2.a.a, 196.2.a.c, 392.2.a.a, 392.2.a.e, 392.2.a.g, 784.2.a.a, 784.2.a.c, 784.2.a.g, 784.2.a.h, 784.2.a.j, 784.2.a.k$^{2}$, 784.2.a.l, 784.2.a.m, 1568.2.a.l, 1568.2.a.q, 1568.2.a.r, 1568.2.a.w, 3136.2.a.bc$^{2}$, 3136.2.a.bg, 3136.2.a.bi, 3136.2.a.bj, 3136.2.a.bm$^{3}$, 3136.2.a.bn, 3136.2.a.bp$^{2}$, 3136.2.a.br, 3136.2.a.bs$^{3}$, 3136.2.a.bu, 3136.2.a.bz, 3136.2.a.j$^{2}$, 3136.2.a.k, 3136.2.a.m, 3136.2.a.s$^{2}$, 3136.2.a.v |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=3,11,23,31,67$, 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 |
|---|---|---|---|---|---|---|
| 56.48.1.hl.1 | $56$ | $21$ | $21$ | $1$ | $1$ | $1^{16}\cdot2^{26}\cdot4$ |
| 56.504.32.cn.1 | $56$ | $2$ | $2$ | $32$ | $8$ | $1^{9}\cdot2^{14}\cdot4$ |
| 56.504.32.er.1 | $56$ | $2$ | $2$ | $32$ | $13$ | $1^{9}\cdot2^{14}\cdot4$ |
| 56.504.34.jg.1 | $56$ | $2$ | $2$ | $34$ | $12$ | $1^{7}\cdot2^{14}\cdot4$ |
| 56.504.34.jx.1 | $56$ | $2$ | $2$ | $34$ | $11$ | $1^{7}\cdot2^{14}\cdot4$ |
| 56.504.37.jx.1 | $56$ | $2$ | $2$ | $37$ | $11$ | $1^{8}\cdot2^{14}$ |
| 56.504.37.lt.1 | $56$ | $2$ | $2$ | $37$ | $12$ | $1^{8}\cdot2^{14}$ |
| 56.504.37.on.1 | $56$ | $2$ | $2$ | $37$ | $18$ | $1^{12}\cdot2^{12}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.2016.145.ewl.1 | $56$ | $2$ | $2$ | $145$ | $57$ | $1^{50}\cdot2^{11}$ |
| 56.2016.145.ewp.1 | $56$ | $2$ | $2$ | $145$ | $46$ | $1^{50}\cdot2^{11}$ |
| 56.2016.145.exr.1 | $56$ | $2$ | $2$ | $145$ | $44$ | $1^{50}\cdot2^{11}$ |
| 56.2016.145.exv.1 | $56$ | $2$ | $2$ | $145$ | $49$ | $1^{50}\cdot2^{11}$ |
| 56.2016.145.gbr.1 | $56$ | $2$ | $2$ | $145$ | $56$ | $1^{50}\cdot2^{11}$ |
| 56.2016.145.gbv.1 | $56$ | $2$ | $2$ | $145$ | $58$ | $1^{50}\cdot2^{11}$ |
| 56.2016.145.gcx.1 | $56$ | $2$ | $2$ | $145$ | $56$ | $1^{50}\cdot2^{11}$ |
| 56.2016.145.gdb.1 | $56$ | $2$ | $2$ | $145$ | $54$ | $1^{50}\cdot2^{11}$ |