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: | $46$ | ||||||
| $\Q$-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: | 56.1344.97.447 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}10&11\\7&22\end{bmatrix}$, $\begin{bmatrix}38&43\\23&18\end{bmatrix}$, $\begin{bmatrix}45&4\\16&39\end{bmatrix}$, $\begin{bmatrix}53&16\\12&31\end{bmatrix}$, $\begin{bmatrix}54&15\\21&50\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 56-isogeny field degree: | $8$ |
| Cyclic 56-torsion field degree: | $192$ |
| Full 56-torsion field degree: | $2304$ |
Jacobian
| Conductor: | $2^{426}\cdot7^{181}$ |
| 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$^{2}$, 98.2.a.b$^{4}$, 112.2.a.a, 112.2.a.b, 196.2.a.a$^{2}$, 196.2.a.b, 196.2.a.c$^{3}$, 392.2.a.a, 392.2.a.b, 392.2.a.c, 392.2.a.d, 392.2.a.e, 392.2.a.f, 392.2.a.g$^{2}$, 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, 784.2.a.g, 784.2.a.h, 784.2.a.k, 784.2.a.l, 784.2.a.m, 3136.2.a.a, 3136.2.a.b, 3136.2.a.bd, 3136.2.a.bg, 3136.2.a.bi, 3136.2.a.bj, 3136.2.a.bk, 3136.2.a.bl, 3136.2.a.bm, 3136.2.a.bn, 3136.2.a.bo, 3136.2.a.bp, 3136.2.a.br, 3136.2.a.bs, 3136.2.a.bu, 3136.2.a.bw, 3136.2.a.bz, 3136.2.a.c, 3136.2.a.f, 3136.2.a.h, 3136.2.a.k, 3136.2.a.m$^{2}$, 3136.2.a.p, 3136.2.a.t, 3136.2.a.u, 3136.2.a.y, 3136.2.a.z |
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 |
|---|---|---|---|---|---|---|
| 28.672.45.bp.1 | $28$ | $2$ | $2$ | $45$ | $12$ | $1^{16}\cdot2^{16}\cdot4$ |
| 56.48.1.jf.1 | $56$ | $28$ | $28$ | $1$ | $1$ | $1^{36}\cdot2^{28}\cdot4$ |
| 56.672.45.ll.1 | $56$ | $2$ | $2$ | $45$ | $24$ | $1^{16}\cdot2^{16}\cdot4$ |
| 56.672.45.zl.1 | $56$ | $2$ | $2$ | $45$ | $24$ | $1^{20}\cdot2^{14}\cdot4$ |
| 56.672.45.bbj.1 | $56$ | $2$ | $2$ | $45$ | $26$ | $1^{20}\cdot2^{14}\cdot4$ |
| 56.672.49.kf.1 | $56$ | $2$ | $2$ | $49$ | $20$ | $1^{16}\cdot2^{16}$ |
| 56.672.49.md.1 | $56$ | $2$ | $2$ | $49$ | $26$ | $1^{16}\cdot2^{16}$ |
| 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.eip.1 | $56$ | $2$ | $2$ | $193$ | $72$ | $1^{70}\cdot2^{13}$ |
| 56.2688.193.eit.1 | $56$ | $2$ | $2$ | $193$ | $86$ | $1^{70}\cdot2^{13}$ |
| 56.2688.193.ejv.1 | $56$ | $2$ | $2$ | $193$ | $86$ | $1^{70}\cdot2^{13}$ |
| 56.2688.193.ejz.1 | $56$ | $2$ | $2$ | $193$ | $79$ | $1^{70}\cdot2^{13}$ |
| 56.2688.193.etz.1 | $56$ | $2$ | $2$ | $193$ | $75$ | $1^{70}\cdot2^{13}$ |
| 56.2688.193.eud.1 | $56$ | $2$ | $2$ | $193$ | $90$ | $1^{70}\cdot2^{13}$ |
| 56.2688.193.evf.1 | $56$ | $2$ | $2$ | $193$ | $90$ | $1^{70}\cdot2^{13}$ |
| 56.2688.193.evj.1 | $56$ | $2$ | $2$ | $193$ | $90$ | $1^{70}\cdot2^{13}$ |
| 56.2688.201.hz.1 | $56$ | $2$ | $2$ | $201$ | $92$ | $1^{32}\cdot2^{32}\cdot4^{2}$ |
| 56.2688.201.ia.1 | $56$ | $2$ | $2$ | $201$ | $80$ | $1^{32}\cdot2^{32}\cdot4^{2}$ |
| 56.2688.201.ib.1 | $56$ | $2$ | $2$ | $201$ | $80$ | $1^{64}\cdot2^{20}$ |
| 56.2688.201.ic.1 | $56$ | $2$ | $2$ | $201$ | $96$ | $1^{64}\cdot2^{20}$ |
| 56.4032.289.fmn.1 | $56$ | $3$ | $3$ | $289$ | $126$ | $1^{102}\cdot2^{43}\cdot4$ |