Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
| 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 | $2^{3}\cdot6^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-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: | 56.672.17.570 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}9&3\\28&5\end{bmatrix}$, $\begin{bmatrix}11&53\\4&3\end{bmatrix}$, $\begin{bmatrix}49&18\\52&35\end{bmatrix}$, $\begin{bmatrix}53&3\\52&33\end{bmatrix}$, $\begin{bmatrix}55&48\\24&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.336.17.bt.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: | $2^{60}\cdot7^{30}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{11}\cdot2^{3}$ |
| Newforms: | 14.2.a.a$^{2}$, 98.2.a.b$^{2}$, 196.2.a.b, 196.2.a.c, 448.2.a.g$^{2}$, 3136.2.a.e$^{2}$, 3136.2.a.h, 3136.2.a.n$^{3}$ |
Rational points
This modular curve has no $\Q_p$ points for $p=17,29,37$, 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 |
|---|---|---|---|---|---|---|
| 56.336.9-28.c.1.2 | $56$ | $2$ | $2$ | $9$ | $0$ | $1^{8}$ |
| 56.336.9-28.c.1.17 | $56$ | $2$ | $2$ | $9$ | $0$ | $1^{8}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.1344.41-56.bi.1.25 | $56$ | $2$ | $2$ | $41$ | $6$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.cg.1.8 | $56$ | $2$ | $2$ | $41$ | $7$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.ie.1.14 | $56$ | $2$ | $2$ | $41$ | $13$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.ig.1.8 | $56$ | $2$ | $2$ | $41$ | $13$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.lw.1.8 | $56$ | $2$ | $2$ | $41$ | $5$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.ly.1.8 | $56$ | $2$ | $2$ | $41$ | $11$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.mr.1.8 | $56$ | $2$ | $2$ | $41$ | $15$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.mt.1.8 | $56$ | $2$ | $2$ | $41$ | $10$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.ok.1.14 | $56$ | $2$ | $2$ | $41$ | $7$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.ok.1.16 | $56$ | $2$ | $2$ | $41$ | $7$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.ol.1.23 | $56$ | $2$ | $2$ | $41$ | $6$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.ol.1.24 | $56$ | $2$ | $2$ | $41$ | $6$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.oo.1.15 | $56$ | $2$ | $2$ | $41$ | $11$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.oo.1.16 | $56$ | $2$ | $2$ | $41$ | $11$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.op.1.14 | $56$ | $2$ | $2$ | $41$ | $5$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.op.1.16 | $56$ | $2$ | $2$ | $41$ | $5$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.pe.1.14 | $56$ | $2$ | $2$ | $41$ | $15$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.pe.1.16 | $56$ | $2$ | $2$ | $41$ | $15$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.pf.1.15 | $56$ | $2$ | $2$ | $41$ | $10$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.pf.1.16 | $56$ | $2$ | $2$ | $41$ | $10$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.pi.1.15 | $56$ | $2$ | $2$ | $41$ | $13$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.pi.1.16 | $56$ | $2$ | $2$ | $41$ | $13$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.pj.1.14 | $56$ | $2$ | $2$ | $41$ | $13$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.pj.1.16 | $56$ | $2$ | $2$ | $41$ | $13$ | $1^{16}\cdot2^{4}$ |
| 56.2016.49-56.x.1.16 | $56$ | $3$ | $3$ | $49$ | $9$ | $1^{20}\cdot2^{6}$ |