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: | $3$ | ||||||
| $\Q$-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: | 56.1008.34.725 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}33&4\\44&15\end{bmatrix}$, $\begin{bmatrix}37&8\\4&43\end{bmatrix}$, $\begin{bmatrix}47&8\\18&53\end{bmatrix}$, $\begin{bmatrix}47&30\\50&37\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.504.34.bm.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: | $2^{139}\cdot7^{68}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{18}\cdot2^{8}$ |
| Newforms: | 49.2.a.a, 98.2.a.a, 98.2.a.b$^{2}$, 196.2.a.a, 196.2.a.b, 196.2.a.c, 392.2.a.a, 392.2.a.b, 392.2.a.d, 392.2.a.e, 392.2.a.h, 3136.2.a.bb, 3136.2.a.bc, 3136.2.a.bm, 3136.2.a.bp, 3136.2.a.bs, 3136.2.a.bt, 3136.2.a.e, 3136.2.a.i, 3136.2.a.j, 3136.2.a.n, 3136.2.a.q, 3136.2.a.s, 3136.2.a.v, 3136.2.a.w |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=3,11,47,67$, 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.g.1.5 | $28$ | $2$ | $2$ | $16$ | $3$ | $1^{10}\cdot2^{4}$ |
| 56.504.16-56.b.1.6 | $56$ | $2$ | $2$ | $16$ | $0$ | $1^{14}\cdot2^{2}$ |
| 56.504.16-56.b.1.16 | $56$ | $2$ | $2$ | $16$ | $0$ | $1^{14}\cdot2^{2}$ |
| 56.504.16-28.g.1.1 | $56$ | $2$ | $2$ | $16$ | $3$ | $1^{10}\cdot2^{4}$ |
| 56.504.16-56.h.1.2 | $56$ | $2$ | $2$ | $16$ | $0$ | $1^{10}\cdot2^{4}$ |
| 56.504.16-56.h.1.16 | $56$ | $2$ | $2$ | $16$ | $0$ | $1^{10}\cdot2^{4}$ |
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.o.1.10 | $56$ | $2$ | $2$ | $67$ | $23$ | $1^{19}\cdot2^{7}$ |
| 56.2016.67-56.p.1.1 | $56$ | $2$ | $2$ | $67$ | $16$ | $1^{19}\cdot2^{7}$ |
| 56.2016.67-56.bc.1.3 | $56$ | $2$ | $2$ | $67$ | $13$ | $1^{19}\cdot2^{7}$ |
| 56.2016.67-56.bf.1.8 | $56$ | $2$ | $2$ | $67$ | $16$ | $1^{19}\cdot2^{7}$ |
| 56.2016.67-56.dg.1.5 | $56$ | $2$ | $2$ | $67$ | $15$ | $1^{19}\cdot2^{7}$ |
| 56.2016.67-56.di.1.4 | $56$ | $2$ | $2$ | $67$ | $13$ | $1^{19}\cdot2^{7}$ |
| 56.2016.67-56.dr.1.2 | $56$ | $2$ | $2$ | $67$ | $16$ | $1^{19}\cdot2^{7}$ |
| 56.2016.67-56.dv.1.3 | $56$ | $2$ | $2$ | $67$ | $16$ | $1^{19}\cdot2^{7}$ |