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 | $6^{2}\cdot12$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $30$ | ||||||
| $\Q$-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: | 56.1008.73.570 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}3&2\\26&25\end{bmatrix}$, $\begin{bmatrix}5&34\\2&23\end{bmatrix}$, $\begin{bmatrix}15&32\\48&7\end{bmatrix}$, $\begin{bmatrix}31&28\\0&31\end{bmatrix}$, $\begin{bmatrix}43&8\\12&55\end{bmatrix}$, $\begin{bmatrix}49&26\\46&11\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 56.2016.73-56.c.1.1, 56.2016.73-56.c.1.2, 56.2016.73-56.c.1.3, 56.2016.73-56.c.1.4, 56.2016.73-56.c.1.5, 56.2016.73-56.c.1.6, 56.2016.73-56.c.1.7, 56.2016.73-56.c.1.8, 56.2016.73-56.c.1.9, 56.2016.73-56.c.1.10, 56.2016.73-56.c.1.11, 56.2016.73-56.c.1.12 |
| Cyclic 56-isogeny field degree: | $32$ |
| Cyclic 56-torsion field degree: | $768$ |
| Full 56-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{352}\cdot7^{146}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{17}\cdot2^{26}\cdot4$ |
| Newforms: | 98.2.a.b$^{2}$, 196.2.a.b, 196.2.a.c, 784.2.a.a$^{3}$, 784.2.a.d$^{3}$, 784.2.a.h$^{3}$, 784.2.a.k$^{3}$, 784.2.a.l$^{3}$, 784.2.a.m$^{3}$, 3136.2.a.a, 3136.2.a.bb, 3136.2.a.be, 3136.2.a.bg, 3136.2.a.bi, 3136.2.a.bj, 3136.2.a.bk, 3136.2.a.bl, 3136.2.a.bm, 3136.2.a.bn, 3136.2.a.bo, 3136.2.a.bp, 3136.2.a.br, 3136.2.a.bs, 3136.2.a.bu, 3136.2.a.bx, 3136.2.a.bz, 3136.2.a.i, 3136.2.a.k, 3136.2.a.m, 3136.2.a.t, 3136.2.a.v |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=5,23,31,37,149$, 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 |
|---|---|---|---|---|---|---|
| 28.504.34.a.1 | $28$ | $2$ | $2$ | $34$ | $12$ | $1^{7}\cdot2^{14}\cdot4$ |
| 56.48.1.a.1 | $56$ | $21$ | $21$ | $1$ | $1$ | $1^{16}\cdot2^{26}\cdot4$ |
| 56.504.32.m.1 | $56$ | $2$ | $2$ | $32$ | $15$ | $1^{9}\cdot2^{14}\cdot4$ |
| 56.504.32.cc.1 | $56$ | $2$ | $2$ | $32$ | $18$ | $1^{9}\cdot2^{14}\cdot4$ |
| 56.504.34.ca.1 | $56$ | $2$ | $2$ | $34$ | $13$ | $1^{7}\cdot2^{14}\cdot4$ |
| 56.504.37.f.1 | $56$ | $2$ | $2$ | $37$ | $13$ | $1^{12}\cdot2^{12}$ |
| 56.504.37.ke.1 | $56$ | $2$ | $2$ | $37$ | $16$ | $1^{8}\cdot2^{14}$ |
| 56.504.37.lu.1 | $56$ | $2$ | $2$ | $37$ | $19$ | $1^{8}\cdot2^{14}$ |
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.c.1 | $56$ | $2$ | $2$ | $145$ | $59$ | $1^{50}\cdot2^{11}$ |
| 56.2016.145.g.1 | $56$ | $2$ | $2$ | $145$ | $69$ | $1^{50}\cdot2^{11}$ |
| 56.2016.145.k.1 | $56$ | $2$ | $2$ | $145$ | $62$ | $1^{50}\cdot2^{11}$ |
| 56.2016.145.q.1 | $56$ | $2$ | $2$ | $145$ | $58$ | $1^{50}\cdot2^{11}$ |
| 56.2016.145.u.1 | $56$ | $2$ | $2$ | $145$ | $60$ | $1^{50}\cdot2^{11}$ |
| 56.2016.145.y.1 | $56$ | $2$ | $2$ | $145$ | $56$ | $1^{50}\cdot2^{11}$ |
| 56.2016.145.bc.1 | $56$ | $2$ | $2$ | $145$ | $48$ | $1^{50}\cdot2^{11}$ |
| 56.2016.145.bg.1 | $56$ | $2$ | $2$ | $145$ | $64$ | $1^{50}\cdot2^{11}$ |
| 56.2016.151.c.1 | $56$ | $2$ | $2$ | $151$ | $65$ | $1^{14}\cdot2^{28}\cdot4^{2}$ |
| 56.2016.151.e.1 | $56$ | $2$ | $2$ | $151$ | $57$ | $1^{14}\cdot2^{28}\cdot4^{2}$ |
| 56.2016.151.l.1 | $56$ | $2$ | $2$ | $151$ | $57$ | $1^{46}\cdot2^{16}$ |
| 56.2016.151.m.1 | $56$ | $2$ | $2$ | $151$ | $69$ | $1^{46}\cdot2^{16}$ |