Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
| Index: | $1008$ | $\PSL_2$-index: | $504$ | ||||
| Genus: | $31 = 1 + \frac{ 504 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (none of which are rational) | Cusp widths | $7^{12}\cdot14^{6}\cdot56^{6}$ | Cusp orbits | $6^{2}\cdot12$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $18$ | ||||||
| $\Q$-gonality: | $10 \le \gamma \le 16$ | ||||||
| $\overline{\Q}$-gonality: | $10 \le \gamma \le 16$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.1008.31.2207 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}13&48\\48&45\end{bmatrix}$, $\begin{bmatrix}27&50\\8&9\end{bmatrix}$, $\begin{bmatrix}31&43\\8&25\end{bmatrix}$, $\begin{bmatrix}55&12\\20&43\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.504.31.ot.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $16$ |
| Cyclic 56-torsion field degree: | $192$ |
| Full 56-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{136}\cdot7^{62}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{17}\cdot2^{7}$ |
| Newforms: | 98.2.a.b$^{2}$, 196.2.a.b, 196.2.a.c, 784.2.a.b, 784.2.a.c, 784.2.a.e, 784.2.a.f, 784.2.a.g, 784.2.a.i, 784.2.a.j, 784.2.a.n, 3136.2.a.b, 3136.2.a.bk, 3136.2.a.bn, 3136.2.a.br, 3136.2.a.h, 3136.2.a.k, 3136.2.a.o$^{3}$, 3136.2.a.u, 3136.2.a.z$^{2}$ |
Rational points
This modular curve has no $\Q_p$ points for $p=3,5,29,37$, 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 |
|---|---|---|---|---|---|---|
| 56.504.13-56.be.1.8 | $56$ | $2$ | $2$ | $13$ | $6$ | $1^{10}\cdot2^{4}$ |
| 56.504.13-56.be.1.16 | $56$ | $2$ | $2$ | $13$ | $6$ | $1^{10}\cdot2^{4}$ |
| 56.504.16-56.cn.1.11 | $56$ | $2$ | $2$ | $16$ | $3$ | $1^{9}\cdot2^{3}$ |
| 56.504.16-56.cn.1.25 | $56$ | $2$ | $2$ | $16$ | $3$ | $1^{9}\cdot2^{3}$ |
| 56.504.16-56.cx.1.7 | $56$ | $2$ | $2$ | $16$ | $9$ | $1^{13}\cdot2$ |
| 56.504.16-56.cx.1.13 | $56$ | $2$ | $2$ | $16$ | $9$ | $1^{13}\cdot2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.2016.67-56.hl.1.11 | $56$ | $2$ | $2$ | $67$ | $31$ | $1^{20}\cdot2^{8}$ |
| 56.2016.67-56.jt.1.11 | $56$ | $2$ | $2$ | $67$ | $31$ | $1^{20}\cdot2^{8}$ |
| 56.2016.67-56.rq.1.3 | $56$ | $2$ | $2$ | $67$ | $34$ | $1^{20}\cdot2^{8}$ |
| 56.2016.67-56.rs.1.7 | $56$ | $2$ | $2$ | $67$ | $34$ | $1^{20}\cdot2^{8}$ |
| 56.2016.67-56.wd.1.6 | $56$ | $2$ | $2$ | $67$ | $28$ | $1^{20}\cdot2^{8}$ |
| 56.2016.67-56.wh.1.6 | $56$ | $2$ | $2$ | $67$ | $28$ | $1^{20}\cdot2^{8}$ |
| 56.2016.67-56.ww.1.6 | $56$ | $2$ | $2$ | $67$ | $35$ | $1^{20}\cdot2^{8}$ |
| 56.2016.67-56.xa.1.6 | $56$ | $2$ | $2$ | $67$ | $35$ | $1^{20}\cdot2^{8}$ |