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 | $2^{4}\cdot6^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $35$ | ||||||
| $\Q$-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: | 56.1344.97.336 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}5&29\\36&23\end{bmatrix}$, $\begin{bmatrix}9&40\\0&45\end{bmatrix}$, $\begin{bmatrix}11&19\\28&5\end{bmatrix}$, $\begin{bmatrix}27&52\\0&15\end{bmatrix}$, $\begin{bmatrix}43&39\\0&13\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 56.2688.97-56.yi.1.1, 56.2688.97-56.yi.1.2, 56.2688.97-56.yi.1.3, 56.2688.97-56.yi.1.4, 56.2688.97-56.yi.1.5, 56.2688.97-56.yi.1.6, 56.2688.97-56.yi.1.7, 56.2688.97-56.yi.1.8 |
| Cyclic 56-isogeny field degree: | $4$ |
| Cyclic 56-torsion field degree: | $96$ |
| Full 56-torsion field degree: | $2304$ |
Jacobian
| Conductor: | $2^{447}\cdot7^{183}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{37}\cdot2^{28}\cdot4$ |
| Newforms: | 14.2.a.a$^{3}$, 56.2.a.a, 56.2.a.b, 98.2.a.b$^{3}$, 112.2.a.a$^{2}$, 112.2.a.b$^{2}$, 112.2.a.c$^{2}$, 196.2.a.b$^{2}$, 196.2.a.c$^{2}$, 392.2.a.c, 392.2.a.f, 392.2.a.g, 784.2.a.a$^{2}$, 784.2.a.d$^{2}$, 784.2.a.h$^{2}$, 784.2.a.k$^{2}$, 784.2.a.l$^{2}$, 784.2.a.m$^{2}$, 3136.2.a.a$^{2}$, 3136.2.a.be, 3136.2.a.bf, 3136.2.a.bg, 3136.2.a.bi, 3136.2.a.bj, 3136.2.a.bl, 3136.2.a.bm$^{2}$, 3136.2.a.bo, 3136.2.a.bp$^{2}$, 3136.2.a.bs$^{2}$, 3136.2.a.bu, 3136.2.a.bx, 3136.2.a.by, 3136.2.a.bz, 3136.2.a.c$^{2}$, 3136.2.a.f, 3136.2.a.k$^{2}$, 3136.2.a.m$^{2}$, 3136.2.a.p$^{2}$, 3136.2.a.t$^{2}$, 3136.2.a.y, 3136.2.a.z$^{2}$ |
Rational points
This modular curve has no $\Q_p$ points for $p=71$, 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 |
|---|---|---|---|---|---|---|
| 28.672.45.t.1 | $28$ | $2$ | $2$ | $45$ | $11$ | $1^{16}\cdot2^{16}\cdot4$ |
| 56.48.1.dg.1 | $56$ | $28$ | $28$ | $1$ | $1$ | $1^{36}\cdot2^{28}\cdot4$ |
| 56.672.45.dm.1 | $56$ | $2$ | $2$ | $45$ | $17$ | $1^{16}\cdot2^{16}\cdot4$ |
| 56.672.45.wq.1 | $56$ | $2$ | $2$ | $45$ | $20$ | $1^{20}\cdot2^{14}\cdot4$ |
| 56.672.45.wr.1 | $56$ | $2$ | $2$ | $45$ | $16$ | $1^{20}\cdot2^{14}\cdot4$ |
| 56.672.49.bz.1 | $56$ | $2$ | $2$ | $49$ | $17$ | $1^{24}\cdot2^{12}$ |
| 56.672.49.ke.1 | $56$ | $2$ | $2$ | $49$ | $24$ | $1^{16}\cdot2^{16}$ |
| 56.672.49.kf.1 | $56$ | $2$ | $2$ | $49$ | $20$ | $1^{16}\cdot2^{16}$ |
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.cpo.1 | $56$ | $2$ | $2$ | $193$ | $75$ | $1^{70}\cdot2^{13}$ |
| 56.2688.193.cpu.1 | $56$ | $2$ | $2$ | $193$ | $67$ | $1^{70}\cdot2^{13}$ |
| 56.2688.193.cqy.1 | $56$ | $2$ | $2$ | $193$ | $65$ | $1^{70}\cdot2^{13}$ |
| 56.2688.193.crc.1 | $56$ | $2$ | $2$ | $193$ | $72$ | $1^{70}\cdot2^{13}$ |
| 56.2688.193.cse.1 | $56$ | $2$ | $2$ | $193$ | $71$ | $1^{70}\cdot2^{13}$ |
| 56.2688.193.csi.1 | $56$ | $2$ | $2$ | $193$ | $81$ | $1^{70}\cdot2^{13}$ |
| 56.2688.193.cto.1 | $56$ | $2$ | $2$ | $193$ | $76$ | $1^{70}\cdot2^{13}$ |
| 56.2688.193.cts.1 | $56$ | $2$ | $2$ | $193$ | $73$ | $1^{70}\cdot2^{13}$ |
| 56.4032.289.coy.1 | $56$ | $3$ | $3$ | $289$ | $113$ | $1^{102}\cdot2^{43}\cdot4$ |