Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $784$ | ||
| Index: | $672$ | $\PSL_2$-index: | $336$ | ||||
| Genus: | $21 = 1 + \frac{ 336 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $2$ are rational) | Cusp widths | $14^{8}\cdot28^{8}$ | Cusp orbits | $1^{2}\cdot2\cdot3^{2}\cdot6$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $5$ | ||||||
| $\Q$-gonality: | $6 \le \gamma \le 12$ | ||||||
| $\overline{\Q}$-gonality: | $6 \le \gamma \le 12$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 28B21 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.672.21.957 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}11&20\\28&3\end{bmatrix}$, $\begin{bmatrix}17&26\\24&39\end{bmatrix}$, $\begin{bmatrix}27&42\\16&15\end{bmatrix}$, $\begin{bmatrix}39&3\\24&17\end{bmatrix}$, $\begin{bmatrix}51&3\\52&45\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 28.336.21.w.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: | $2^{60}\cdot7^{37}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{13}\cdot2^{4}$ |
| Newforms: | 14.2.a.a$^{2}$, 98.2.a.b$^{2}$, 112.2.a.a, 112.2.a.b, 112.2.a.c, 196.2.a.b, 196.2.a.c, 784.2.a.a, 784.2.a.b, 784.2.a.d, 784.2.a.e, 784.2.a.f, 784.2.a.h, 784.2.a.i, 784.2.a.n |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal 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 | $56$ | $2$ | $2$ | $9$ | $0$ | $1^{10}\cdot2$ |
| 56.336.9-28.c.1.22 | $56$ | $2$ | $2$ | $9$ | $0$ | $1^{10}\cdot2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.1344.41-28.l.1.20 | $56$ | $2$ | $2$ | $41$ | $11$ | $1^{14}\cdot2^{3}$ |
| 56.1344.41-28.bb.1.20 | $56$ | $2$ | $2$ | $41$ | $10$ | $1^{14}\cdot2^{3}$ |
| 56.1344.41-28.bl.1.6 | $56$ | $2$ | $2$ | $41$ | $14$ | $1^{14}\cdot2^{3}$ |
| 56.1344.41-28.bm.1.12 | $56$ | $2$ | $2$ | $41$ | $10$ | $1^{14}\cdot2^{3}$ |
| 56.1344.41-56.cg.1.8 | $56$ | $2$ | $2$ | $41$ | $7$ | $1^{14}\cdot2^{3}$ |
| 56.1344.41-56.hl.1.8 | $56$ | $2$ | $2$ | $41$ | $20$ | $1^{14}\cdot2^{3}$ |
| 56.1344.41-56.kb.1.8 | $56$ | $2$ | $2$ | $41$ | $11$ | $1^{14}\cdot2^{3}$ |
| 56.1344.41-56.ki.1.8 | $56$ | $2$ | $2$ | $41$ | $17$ | $1^{14}\cdot2^{3}$ |
| 56.1344.45-56.ha.1.14 | $56$ | $2$ | $2$ | $45$ | $13$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.ha.1.16 | $56$ | $2$ | $2$ | $45$ | $13$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.hb.1.22 | $56$ | $2$ | $2$ | $45$ | $8$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.hb.1.24 | $56$ | $2$ | $2$ | $45$ | $8$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.hm.1.12 | $56$ | $2$ | $2$ | $45$ | $21$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.hm.1.16 | $56$ | $2$ | $2$ | $45$ | $21$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.hn.1.14 | $56$ | $2$ | $2$ | $45$ | $11$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.hn.1.16 | $56$ | $2$ | $2$ | $45$ | $11$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.hq.1.12 | $56$ | $2$ | $2$ | $45$ | $13$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.hq.1.16 | $56$ | $2$ | $2$ | $45$ | $13$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.hr.1.12 | $56$ | $2$ | $2$ | $45$ | $12$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.hr.1.16 | $56$ | $2$ | $2$ | $45$ | $12$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.hu.1.14 | $56$ | $2$ | $2$ | $45$ | $15$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.hu.1.16 | $56$ | $2$ | $2$ | $45$ | $15$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.hv.1.14 | $56$ | $2$ | $2$ | $45$ | $19$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.hv.1.16 | $56$ | $2$ | $2$ | $45$ | $19$ | $1^{16}\cdot2^{4}$ |
| 56.2016.61-28.be.1.2 | $56$ | $3$ | $3$ | $61$ | $15$ | $1^{26}\cdot2^{7}$ |