Invariants
| Level: | $56$ | $\SL_2$-level: | $28$ | 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 | $14^{12}\cdot28^{12}$ | Cusp orbits | $6^{2}\cdot12$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $7$ | ||||||
| $\Q$-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: | 56.1008.31.187 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}3&46\\10&31\end{bmatrix}$, $\begin{bmatrix}41&32\\32&15\end{bmatrix}$, $\begin{bmatrix}47&20\\50&5\end{bmatrix}$, $\begin{bmatrix}51&6\\22&9\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.504.31.bh.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: | $2^{142}\cdot7^{57}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{17}\cdot2^{7}$ |
| Newforms: | 98.2.a.b$^{2}$, 112.2.a.c$^{2}$, 196.2.a.b, 196.2.a.c, 448.2.a.c, 448.2.a.d, 448.2.a.g, 784.2.a.d, 784.2.a.f$^{3}$, 3136.2.a.b, 3136.2.a.bc, 3136.2.a.bm, 3136.2.a.bp, 3136.2.a.bs, 3136.2.a.bt, 3136.2.a.h, 3136.2.a.j, 3136.2.a.n, 3136.2.a.s, 3136.2.a.u |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=3,5,\ldots,179$, 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 |
|---|---|---|---|---|---|---|
| 28.504.13-28.b.1.2 | $28$ | $2$ | $2$ | $13$ | $3$ | $1^{10}\cdot2^{4}$ |
| 56.504.13-28.b.1.2 | $56$ | $2$ | $2$ | $13$ | $3$ | $1^{10}\cdot2^{4}$ |
| 56.504.16-56.b.1.6 | $56$ | $2$ | $2$ | $16$ | $0$ | $1^{13}\cdot2$ |
| 56.504.16-56.b.1.7 | $56$ | $2$ | $2$ | $16$ | $0$ | $1^{13}\cdot2$ |
| 56.504.16-56.f.1.4 | $56$ | $2$ | $2$ | $16$ | $4$ | $1^{9}\cdot2^{3}$ |
| 56.504.16-56.f.1.8 | $56$ | $2$ | $2$ | $16$ | $4$ | $1^{9}\cdot2^{3}$ |
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.x.1.3 | $56$ | $2$ | $2$ | $67$ | $16$ | $1^{20}\cdot2^{8}$ |
| 56.2016.67-56.x.1.4 | $56$ | $2$ | $2$ | $67$ | $16$ | $1^{20}\cdot2^{8}$ |
| 56.2016.67-56.y.1.9 | $56$ | $2$ | $2$ | $67$ | $25$ | $1^{20}\cdot2^{8}$ |
| 56.2016.67-56.y.1.10 | $56$ | $2$ | $2$ | $67$ | $25$ | $1^{20}\cdot2^{8}$ |
| 56.2016.67-56.dr.1.1 | $56$ | $2$ | $2$ | $67$ | $16$ | $1^{20}\cdot2^{8}$ |
| 56.2016.67-56.dr.1.2 | $56$ | $2$ | $2$ | $67$ | $16$ | $1^{20}\cdot2^{8}$ |
| 56.2016.67-56.ds.1.3 | $56$ | $2$ | $2$ | $67$ | $27$ | $1^{20}\cdot2^{8}$ |
| 56.2016.67-56.ds.1.4 | $56$ | $2$ | $2$ | $67$ | $27$ | $1^{20}\cdot2^{8}$ |