Invariants
| Level: | $56$ | $\SL_2$-level: | $28$ | Newform level: | $784$ | ||
| 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 | $14^{24}$ | Cusp orbits | $2\cdot3^{2}\cdot4\cdot6^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-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: | 14B17 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.672.17.452 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}3&50\\14&11\end{bmatrix}$, $\begin{bmatrix}27&49\\2&43\end{bmatrix}$, $\begin{bmatrix}39&27\\48&41\end{bmatrix}$, $\begin{bmatrix}47&30\\2&23\end{bmatrix}$, $\begin{bmatrix}47&37\\44&37\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 28.336.17.d.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $8$ |
| Cyclic 56-torsion field degree: | $192$ |
| Full 56-torsion field degree: | $4608$ |
Jacobian
| Conductor: | $2^{45}\cdot7^{30}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{11}\cdot2^{3}$ |
| Newforms: | 14.2.a.a$^{2}$, 49.2.a.a$^{2}$, 98.2.a.a, 98.2.a.b, 112.2.a.c$^{2}$, 784.2.a.b, 784.2.a.d$^{2}$, 784.2.a.f, 784.2.a.l, 784.2.a.m |
Rational points
This modular curve has no $\Q_p$ points for $p=5,17$, 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.a.1.5 | $56$ | $2$ | $2$ | $9$ | $0$ | $1^{8}$ |
| 56.336.9-28.a.1.6 | $56$ | $2$ | $2$ | $9$ | $0$ | $1^{8}$ |
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.d.1.1 | $56$ | $2$ | $2$ | $37$ | $2$ | $2^{2}\cdot4^{2}\cdot8$ |
| 56.1344.37-28.d.1.7 | $56$ | $2$ | $2$ | $37$ | $2$ | $2^{2}\cdot4^{2}\cdot8$ |
| 56.1344.37-56.d.1.1 | $56$ | $2$ | $2$ | $37$ | $2$ | $2^{2}\cdot4^{2}\cdot8$ |
| 56.1344.37-56.d.1.15 | $56$ | $2$ | $2$ | $37$ | $2$ | $2^{2}\cdot4^{2}\cdot8$ |
| 56.1344.41-28.r.1.1 | $56$ | $2$ | $2$ | $41$ | $7$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-28.w.1.1 | $56$ | $2$ | $2$ | $41$ | $11$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-28.z.1.1 | $56$ | $2$ | $2$ | $41$ | $7$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-28.bb.1.2 | $56$ | $2$ | $2$ | $41$ | $10$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.eq.1.4 | $56$ | $2$ | $2$ | $41$ | $14$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.fz.1.3 | $56$ | $2$ | $2$ | $41$ | $17$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.gu.1.4 | $56$ | $2$ | $2$ | $41$ | $10$ | $1^{16}\cdot2^{4}$ |
| 56.1344.41-56.hh.1.4 | $56$ | $2$ | $2$ | $41$ | $12$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-28.s.1.4 | $56$ | $2$ | $2$ | $45$ | $4$ | $2^{6}\cdot4^{2}\cdot8$ |
| 56.1344.45-28.s.1.6 | $56$ | $2$ | $2$ | $45$ | $4$ | $2^{6}\cdot4^{2}\cdot8$ |
| 56.1344.45-56.dr.1.2 | $56$ | $2$ | $2$ | $45$ | $6$ | $2^{6}\cdot4^{2}\cdot8$ |
| 56.1344.45-56.dr.1.16 | $56$ | $2$ | $2$ | $45$ | $6$ | $2^{6}\cdot4^{2}\cdot8$ |
| 56.2016.49-28.b.1.1 | $56$ | $3$ | $3$ | $49$ | $9$ | $1^{20}\cdot2^{6}$ |