Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
| Index: | $1008$ | $\PSL_2$-index: | $504$ | ||||
| Genus: | $34 = 1 + \frac{ 504 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 18 }{2}$ | ||||||
| Cusps: | $18$ (none of which are rational) | Cusp widths | $14^{12}\cdot56^{6}$ | Cusp orbits | $6^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $14$ | ||||||
| $\Q$-gonality: | $9 \le \gamma \le 16$ | ||||||
| $\overline{\Q}$-gonality: | $9 \le \gamma \le 16$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.1008.34.2395 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}20&23\\27&20\end{bmatrix}$, $\begin{bmatrix}24&39\\5&38\end{bmatrix}$, $\begin{bmatrix}32&7\\35&24\end{bmatrix}$, $\begin{bmatrix}55&8\\32&43\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.504.34.fp.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: | $2^{138}\cdot7^{68}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{10}\cdot2^{12}$ |
| Newforms: | 98.2.a.b$^{3}$, 196.2.a.a, 196.2.a.b, 196.2.a.c$^{2}$, 392.2.a.a, 392.2.a.e, 392.2.a.g, 3136.2.a.a, 3136.2.a.b, 3136.2.a.bk, 3136.2.a.bm, 3136.2.a.bn, 3136.2.a.bp, 3136.2.a.br, 3136.2.a.bs, 3136.2.a.h, 3136.2.a.k, 3136.2.a.t, 3136.2.a.u |
Rational points
This modular curve has no $\Q_p$ points for $p=11,67$, 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 |
|---|---|---|---|---|---|---|
| 56.48.0-56.bn.1.1 | $56$ | $21$ | $21$ | $0$ | $0$ | full Jacobian |
| 56.504.16-28.s.1.4 | $56$ | $2$ | $2$ | $16$ | $2$ | $1^{6}\cdot2^{6}$ |
| 56.504.16-28.s.1.6 | $56$ | $2$ | $2$ | $16$ | $2$ | $1^{6}\cdot2^{6}$ |
| 56.504.16-56.ck.1.4 | $56$ | $2$ | $2$ | $16$ | $3$ | $1^{6}\cdot2^{6}$ |
| 56.504.16-56.ck.1.28 | $56$ | $2$ | $2$ | $16$ | $3$ | $1^{6}\cdot2^{6}$ |
| 56.504.16-56.cx.1.7 | $56$ | $2$ | $2$ | $16$ | $9$ | $1^{6}\cdot2^{6}$ |
| 56.504.16-56.cx.1.31 | $56$ | $2$ | $2$ | $16$ | $9$ | $1^{6}\cdot2^{6}$ |
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.wb.1.4 | $56$ | $2$ | $2$ | $67$ | $20$ | $1^{27}\cdot2^{3}$ |
| 56.2016.67-56.wd.1.6 | $56$ | $2$ | $2$ | $67$ | $28$ | $1^{27}\cdot2^{3}$ |
| 56.2016.67-56.wm.1.2 | $56$ | $2$ | $2$ | $67$ | $37$ | $1^{27}\cdot2^{3}$ |
| 56.2016.67-56.wo.1.5 | $56$ | $2$ | $2$ | $67$ | $25$ | $1^{27}\cdot2^{3}$ |
| 56.2016.67-56.yx.1.4 | $56$ | $2$ | $2$ | $67$ | $19$ | $1^{27}\cdot2^{3}$ |
| 56.2016.67-56.yz.1.6 | $56$ | $2$ | $2$ | $67$ | $24$ | $1^{27}\cdot2^{3}$ |
| 56.2016.67-56.zj.1.2 | $56$ | $2$ | $2$ | $67$ | $33$ | $1^{27}\cdot2^{3}$ |
| 56.2016.67-56.zl.1.3 | $56$ | $2$ | $2$ | $67$ | $30$ | $1^{27}\cdot2^{3}$ |