Invariants
| Level: | $56$ | $\SL_2$-level: | $28$ | 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 | $28^{18}$ | Cusp orbits | $6^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $8$ | ||||||
| $\Q$-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: | 56.1008.34.381 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}1&4\\54&15\end{bmatrix}$, $\begin{bmatrix}21&2\\36&21\end{bmatrix}$, $\begin{bmatrix}23&32\\44&5\end{bmatrix}$, $\begin{bmatrix}37&52\\40&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.504.34.j.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $32$ |
| Cyclic 56-torsion field degree: | $384$ |
| 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.bb, 3136.2.a.bc, 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.i, 3136.2.a.j, 3136.2.a.s, 3136.2.a.v |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=3,11,31,67$, 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 |
|---|---|---|---|---|---|---|
| 28.504.16-28.d.1.7 | $28$ | $2$ | $2$ | $16$ | $2$ | $1^{6}\cdot2^{6}$ |
| 56.48.0-56.g.1.5 | $56$ | $21$ | $21$ | $0$ | $0$ | full Jacobian |
| 56.504.16-56.b.1.6 | $56$ | $2$ | $2$ | $16$ | $0$ | $1^{6}\cdot2^{6}$ |
| 56.504.16-56.b.1.13 | $56$ | $2$ | $2$ | $16$ | $0$ | $1^{6}\cdot2^{6}$ |
| 56.504.16-56.c.1.2 | $56$ | $2$ | $2$ | $16$ | $6$ | $1^{6}\cdot2^{6}$ |
| 56.504.16-56.c.1.11 | $56$ | $2$ | $2$ | $16$ | $6$ | $1^{6}\cdot2^{6}$ |
| 56.504.16-28.d.1.4 | $56$ | $2$ | $2$ | $16$ | $2$ | $1^{6}\cdot2^{6}$ |
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.by.1.5 | $56$ | $2$ | $2$ | $67$ | $23$ | $1^{27}\cdot2^{3}$ |
| 56.2016.67-56.cb.1.7 | $56$ | $2$ | $2$ | $67$ | $25$ | $1^{27}\cdot2^{3}$ |
| 56.2016.67-56.ck.1.1 | $56$ | $2$ | $2$ | $67$ | $22$ | $1^{27}\cdot2^{3}$ |
| 56.2016.67-56.cn.1.2 | $56$ | $2$ | $2$ | $67$ | $16$ | $1^{27}\cdot2^{3}$ |
| 56.2016.67-56.dg.1.5 | $56$ | $2$ | $2$ | $67$ | $15$ | $1^{27}\cdot2^{3}$ |
| 56.2016.67-56.dj.1.5 | $56$ | $2$ | $2$ | $67$ | $24$ | $1^{27}\cdot2^{3}$ |
| 56.2016.67-56.ds.1.3 | $56$ | $2$ | $2$ | $67$ | $27$ | $1^{27}\cdot2^{3}$ |
| 56.2016.67-56.dv.1.1 | $56$ | $2$ | $2$ | $67$ | $16$ | $1^{27}\cdot2^{3}$ |