Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $196$ | ||
| Index: | $672$ | $\PSL_2$-index: | $336$ | ||||
| Genus: | $17 = 1 + \frac{ 336 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (none of which are rational) | Cusp widths | $7^{16}\cdot28^{8}$ | Cusp orbits | $2^{3}\cdot3^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $7 \le \gamma \le 12$ | ||||||
| $\overline{\Q}$-gonality: | $7 \le \gamma \le 12$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 28A17 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.672.17.469 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}9&47\\12&27\end{bmatrix}$, $\begin{bmatrix}27&44\\28&29\end{bmatrix}$, $\begin{bmatrix}31&22\\28&39\end{bmatrix}$, $\begin{bmatrix}33&42\\28&5\end{bmatrix}$, $\begin{bmatrix}43&19\\28&13\end{bmatrix}$, $\begin{bmatrix}51&12\\40&41\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 28.336.17.m.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $4$ |
| Cyclic 56-torsion field degree: | $96$ |
| Full 56-torsion field degree: | $4608$ |
Jacobian
| Conductor: | $2^{18}\cdot7^{30}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{11}\cdot2^{3}$ |
| Newforms: | 14.2.a.a$^{4}$, 49.2.a.a$^{3}$, 98.2.a.a$^{2}$, 98.2.a.b$^{2}$, 196.2.a.b$^{2}$, 196.2.a.c |
Rational points
This modular curve has no $\Q_p$ points for $p=5,17$, and therefore no rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 7.56.1.b.1 | $7$ | $12$ | $6$ | $1$ | $0$ | $1^{10}\cdot2^{3}$ |
| 8.12.0-4.c.1.6 | $8$ | $56$ | $56$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.336.9-28.c.1.2 | $56$ | $2$ | $2$ | $9$ | $0$ | $1^{8}$ |
| 56.336.9-28.c.1.4 | $56$ | $2$ | $2$ | $9$ | $0$ | $1^{8}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.1344.37-28.f.1.3 | $56$ | $2$ | $2$ | $37$ | $0$ | $2^{2}\cdot4^{2}\cdot8$ |
| 56.1344.37-28.f.1.15 | $56$ | $2$ | $2$ | $37$ | $0$ | $2^{2}\cdot4^{2}\cdot8$ |
| 56.1344.37-28.f.2.3 | $56$ | $2$ | $2$ | $37$ | $0$ | $2^{2}\cdot4^{2}\cdot8$ |
| 56.1344.37-28.f.2.6 | $56$ | $2$ | $2$ | $37$ | $0$ | $2^{2}\cdot4^{2}\cdot8$ |
| 56.1344.37-56.k.1.31 | $56$ | $2$ | $2$ | $37$ | $0$ | $2^{2}\cdot4^{2}\cdot8$ |
| 56.1344.37-56.k.1.32 | $56$ | $2$ | $2$ | $37$ | $0$ | $2^{2}\cdot4^{2}\cdot8$ |
| 56.1344.37-56.l.1.30 | $56$ | $2$ | $2$ | $37$ | $0$ | $2^{2}\cdot4^{2}\cdot8$ |
| 56.1344.37-56.l.1.32 | $56$ | $2$ | $2$ | $37$ | $0$ | $2^{2}\cdot4^{2}\cdot8$ |
| 56.1344.37-56.n.1.16 | $56$ | $2$ | $2$ | $37$ | $0$ | $2^{2}\cdot4^{2}\cdot8$ |
| 56.1344.37-56.n.1.32 | $56$ | $2$ | $2$ | $37$ | $0$ | $2^{2}\cdot4^{2}\cdot8$ |
| 56.1344.37-56.n.2.31 | $56$ | $2$ | $2$ | $37$ | $0$ | $2^{2}\cdot4^{2}\cdot8$ |
| 56.1344.37-56.n.2.32 | $56$ | $2$ | $2$ | $37$ | $0$ | $2^{2}\cdot4^{2}\cdot8$ |
| 56.1344.41-28.j.1.32 | $56$ | $2$ | $2$ | $41$ | $3$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-28.bb.1.20 | $56$ | $2$ | $2$ | $41$ | $10$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-28.bt.1.8 | $56$ | $2$ | $2$ | $41$ | $6$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-28.bu.1.9 | $56$ | $2$ | $2$ | $41$ | $8$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.cc.1.13 | $56$ | $2$ | $2$ | $41$ | $8$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.hk.1.13 | $56$ | $2$ | $2$ | $41$ | $14$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.me.1.14 | $56$ | $2$ | $2$ | $41$ | $16$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.ml.1.13 | $56$ | $2$ | $2$ | $41$ | $7$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.oe.1.12 | $56$ | $2$ | $2$ | $41$ | $8$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.oe.1.24 | $56$ | $2$ | $2$ | $41$ | $8$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.of.1.24 | $56$ | $2$ | $2$ | $41$ | $3$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.of.1.36 | $56$ | $2$ | $2$ | $41$ | $3$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.oq.1.22 | $56$ | $2$ | $2$ | $41$ | $16$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.oq.1.24 | $56$ | $2$ | $2$ | $41$ | $16$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.or.1.20 | $56$ | $2$ | $2$ | $41$ | $6$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.or.1.24 | $56$ | $2$ | $2$ | $41$ | $6$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.ou.1.12 | $56$ | $2$ | $2$ | $41$ | $8$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.ou.1.24 | $56$ | $2$ | $2$ | $41$ | $8$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.ov.1.12 | $56$ | $2$ | $2$ | $41$ | $7$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.ov.1.24 | $56$ | $2$ | $2$ | $41$ | $7$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.oy.1.12 | $56$ | $2$ | $2$ | $41$ | $10$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.oy.1.24 | $56$ | $2$ | $2$ | $41$ | $10$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.oz.1.12 | $56$ | $2$ | $2$ | $41$ | $14$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.oz.1.24 | $56$ | $2$ | $2$ | $41$ | $14$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.go.1.16 | $56$ | $2$ | $2$ | $45$ | $2$ | $2^{6}\cdot4^{2}\cdot8$ |
| 56.1344.45-56.go.1.32 | $56$ | $2$ | $2$ | $45$ | $2$ | $2^{6}\cdot4^{2}\cdot8$ |
| 56.1344.45-56.gp.1.16 | $56$ | $2$ | $2$ | $45$ | $4$ | $2^{6}\cdot4^{2}\cdot8$ |
| 56.1344.45-56.gp.1.32 | $56$ | $2$ | $2$ | $45$ | $4$ | $2^{6}\cdot4^{2}\cdot8$ |
| 56.2016.49-28.g.1.16 | $56$ | $3$ | $3$ | $49$ | $3$ | $1^{20}\cdot2^{6}$ |