Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
| Index: | $1344$ | $\PSL_2$-index: | $1344$ | ||||
| Genus: | $97 = 1 + \frac{ 1344 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 32 }{2}$ | ||||||
| Cusps: | $32$ (none of which are rational) | Cusp widths | $28^{16}\cdot56^{16}$ | Cusp orbits | $2^{2}\cdot4\cdot6^{2}\cdot12$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $47$ | ||||||
| $\Q$-gonality: | $24 \le \gamma \le 48$ | ||||||
| $\overline{\Q}$-gonality: | $24 \le \gamma \le 48$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.1344.97.953 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}13&2\\2&43\end{bmatrix}$, $\begin{bmatrix}31&14\\14&17\end{bmatrix}$, $\begin{bmatrix}38&15\\51&18\end{bmatrix}$, $\begin{bmatrix}42&15\\37&22\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 56-isogeny field degree: | $4$ |
| Cyclic 56-torsion field degree: | $96$ |
| Full 56-torsion field degree: | $2304$ |
Jacobian
| Conductor: | $2^{447}\cdot7^{181}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{37}\cdot2^{28}\cdot4$ |
| Newforms: | 14.2.a.a$^{2}$, 56.2.a.a, 56.2.a.b, 98.2.a.a, 98.2.a.b$^{3}$, 112.2.a.a, 112.2.a.b, 196.2.a.a, 196.2.a.b, 196.2.a.c$^{2}$, 392.2.a.c, 392.2.a.f, 392.2.a.g, 448.2.a.c, 448.2.a.d, 448.2.a.g, 448.2.a.i, 448.2.a.j, 784.2.a.a, 784.2.a.b$^{2}$, 784.2.a.c, 784.2.a.e, 784.2.a.g$^{2}$, 784.2.a.h, 784.2.a.i, 784.2.a.j, 784.2.a.k$^{2}$, 784.2.a.l$^{2}$, 784.2.a.m$^{2}$, 3136.2.a.b, 3136.2.a.bb, 3136.2.a.bd, 3136.2.a.bg, 3136.2.a.bi, 3136.2.a.bj, 3136.2.a.bk$^{2}$, 3136.2.a.bl, 3136.2.a.bn$^{2}$, 3136.2.a.bo, 3136.2.a.br$^{2}$, 3136.2.a.bu, 3136.2.a.bw, 3136.2.a.bz, 3136.2.a.e, 3136.2.a.f, 3136.2.a.h, 3136.2.a.i, 3136.2.a.m$^{2}$, 3136.2.a.q, 3136.2.a.u, 3136.2.a.v, 3136.2.a.w, 3136.2.a.y |
Rational points
This modular curve has real points and $\Q_p$ points for good $p < 8192$, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.48.1.hq.1 | $56$ | $28$ | $28$ | $1$ | $1$ | $1^{36}\cdot2^{28}\cdot4$ |
| 56.672.45.kj.1 | $56$ | $2$ | $2$ | $45$ | $13$ | $1^{16}\cdot2^{16}\cdot4$ |
| 56.672.45.ku.1 | $56$ | $2$ | $2$ | $45$ | $24$ | $1^{16}\cdot2^{16}\cdot4$ |
| 56.672.45.yw.1 | $56$ | $2$ | $2$ | $45$ | $26$ | $1^{20}\cdot2^{14}\cdot4$ |
| 56.672.45.bay.1 | $56$ | $2$ | $2$ | $45$ | $25$ | $1^{20}\cdot2^{14}\cdot4$ |
| 56.672.49.jy.1 | $56$ | $2$ | $2$ | $49$ | $20$ | $1^{16}\cdot2^{16}$ |
| 56.672.49.ls.1 | $56$ | $2$ | $2$ | $49$ | $27$ | $1^{16}\cdot2^{16}$ |
| 56.672.49.oo.1 | $56$ | $2$ | $2$ | $49$ | $26$ | $1^{24}\cdot2^{12}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.2688.193.eck.1 | $56$ | $2$ | $2$ | $193$ | $73$ | $1^{70}\cdot2^{13}$ |
| 56.2688.193.eco.1 | $56$ | $2$ | $2$ | $193$ | $87$ | $1^{70}\cdot2^{13}$ |
| 56.2688.193.edq.1 | $56$ | $2$ | $2$ | $193$ | $87$ | $1^{70}\cdot2^{13}$ |
| 56.2688.193.edu.1 | $56$ | $2$ | $2$ | $193$ | $80$ | $1^{70}\cdot2^{13}$ |
| 56.2688.193.enu.1 | $56$ | $2$ | $2$ | $193$ | $84$ | $1^{70}\cdot2^{13}$ |
| 56.2688.193.eny.1 | $56$ | $2$ | $2$ | $193$ | $99$ | $1^{70}\cdot2^{13}$ |
| 56.2688.193.epa.1 | $56$ | $2$ | $2$ | $193$ | $83$ | $1^{70}\cdot2^{13}$ |
| 56.2688.193.epe.1 | $56$ | $2$ | $2$ | $193$ | $83$ | $1^{70}\cdot2^{13}$ |
| 56.4032.289.fbu.1 | $56$ | $3$ | $3$ | $289$ | $127$ | $1^{102}\cdot2^{43}\cdot4$ |