Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
| Index: | $8064$ | $\PSL_2$-index: | $4032$ | ||||
| Genus: | $277 = 1 + \frac{ 4032 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 120 }{2}$ | ||||||
| Cusps: | $120$ (none of which are rational) | Cusp widths | $14^{48}\cdot28^{24}\cdot56^{48}$ | Cusp orbits | $6^{10}\cdot12^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $41$ | ||||||
| $\Q$-gonality: | $40 \le \gamma \le 84$ | ||||||
| $\overline{\Q}$-gonality: | $40 \le \gamma \le 84$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.8064.277.489 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}15&40\\28&41\end{bmatrix}$, $\begin{bmatrix}17&28\\0&3\end{bmatrix}$, $\begin{bmatrix}27&28\\42&41\end{bmatrix}$, $\begin{bmatrix}29&12\\50&41\end{bmatrix}$, $\begin{bmatrix}55&0\\48&1\end{bmatrix}$ |
| $\GL_2(\Z/56\Z)$-subgroup: | $C_6\times D_4^2$ |
| Contains $-I$: | no $\quad$ (see 56.4032.277.hv.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $4$ |
| Cyclic 56-torsion field degree: | $48$ |
| Full 56-torsion field degree: | $384$ |
Jacobian
Rational points
This modular curve has no $\Q_p$ points for $p=3,5,11,\ldots,409$, and therefore no rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 7.84.1.a.1 | $7$ | $96$ | $48$ | $1$ | $0$ | $1^{78}\cdot2^{35}\cdot4^{8}\cdot6^{8}\cdot12^{4}$ |
| 8.96.0-8.h.1.8 | $8$ | $84$ | $84$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.2688.93-56.ez.1.30 | $56$ | $3$ | $3$ | $93$ | $12$ | $1^{58}\cdot2^{21}\cdot4^{6}\cdot6^{6}\cdot12^{2}$ |
| 56.4032.133-56.cr.1.24 | $56$ | $2$ | $2$ | $133$ | $7$ | $1^{44}\cdot2^{18}\cdot4^{4}\cdot6^{4}\cdot12^{2}$ |
| 56.4032.133-56.cr.1.42 | $56$ | $2$ | $2$ | $133$ | $7$ | $1^{44}\cdot2^{18}\cdot4^{4}\cdot6^{4}\cdot12^{2}$ |
| 56.4032.133-56.cw.1.21 | $56$ | $2$ | $2$ | $133$ | $7$ | $1^{44}\cdot2^{18}\cdot4^{4}\cdot6^{4}\cdot12^{2}$ |
| 56.4032.133-56.cw.1.44 | $56$ | $2$ | $2$ | $133$ | $7$ | $1^{44}\cdot2^{18}\cdot4^{4}\cdot6^{4}\cdot12^{2}$ |
| 56.4032.133-56.eo.1.29 | $56$ | $2$ | $2$ | $133$ | $41$ | $2^{8}\cdot4^{8}\cdot6^{8}\cdot12^{4}$ |
| 56.4032.133-56.eo.1.30 | $56$ | $2$ | $2$ | $133$ | $41$ | $2^{8}\cdot4^{8}\cdot6^{8}\cdot12^{4}$ |
| 56.4032.139-56.hu.1.14 | $56$ | $2$ | $2$ | $139$ | $19$ | $1^{38}\cdot2^{18}\cdot4^{4}\cdot6^{4}\cdot12^{2}$ |
| 56.4032.139-56.hu.1.20 | $56$ | $2$ | $2$ | $139$ | $19$ | $1^{38}\cdot2^{18}\cdot4^{4}\cdot6^{4}\cdot12^{2}$ |
| 56.4032.139-56.hx.1.11 | $56$ | $2$ | $2$ | $139$ | $19$ | $1^{38}\cdot2^{18}\cdot4^{4}\cdot6^{4}\cdot12^{2}$ |
| 56.4032.139-56.hx.1.24 | $56$ | $2$ | $2$ | $139$ | $19$ | $1^{38}\cdot2^{18}\cdot4^{4}\cdot6^{4}\cdot12^{2}$ |
| 56.4032.139-56.nb.1.5 | $56$ | $2$ | $2$ | $139$ | $19$ | $1^{42}\cdot2^{16}\cdot4^{4}\cdot6^{4}\cdot12^{2}$ |
| 56.4032.139-56.nb.1.32 | $56$ | $2$ | $2$ | $139$ | $19$ | $1^{42}\cdot2^{16}\cdot4^{4}\cdot6^{4}\cdot12^{2}$ |
| 56.4032.139-56.nd.1.10 | $56$ | $2$ | $2$ | $139$ | $19$ | $1^{42}\cdot2^{16}\cdot4^{4}\cdot6^{4}\cdot12^{2}$ |
| 56.4032.139-56.nd.1.18 | $56$ | $2$ | $2$ | $139$ | $19$ | $1^{42}\cdot2^{16}\cdot4^{4}\cdot6^{4}\cdot12^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.16128.553-56.if.1.24 | $56$ | $2$ | $2$ | $553$ | $85$ | $1^{96}\cdot2^{30}\cdot4^{12}\cdot6^{8}\cdot12^{2}$ |
| 56.16128.553-56.ik.2.15 | $56$ | $2$ | $2$ | $553$ | $95$ | $1^{96}\cdot2^{30}\cdot4^{12}\cdot6^{8}\cdot12^{2}$ |
| 56.16128.553-56.jg.1.16 | $56$ | $2$ | $2$ | $553$ | $85$ | $1^{96}\cdot2^{30}\cdot4^{12}\cdot6^{8}\cdot12^{2}$ |
| 56.16128.553-56.jl.1.12 | $56$ | $2$ | $2$ | $553$ | $101$ | $1^{96}\cdot2^{30}\cdot4^{12}\cdot6^{8}\cdot12^{2}$ |
| 56.16128.577-56.oc.2.6 | $56$ | $2$ | $2$ | $577$ | $99$ | $1^{60}\cdot2^{52}\cdot4^{10}\cdot6^{8}\cdot12^{4}$ |
| 56.16128.577-56.op.1.4 | $56$ | $2$ | $2$ | $577$ | $106$ | $1^{60}\cdot2^{52}\cdot4^{10}\cdot6^{8}\cdot12^{4}$ |
| 56.16128.577-56.sf.2.1 | $56$ | $2$ | $2$ | $577$ | $99$ | $1^{60}\cdot2^{52}\cdot4^{10}\cdot6^{8}\cdot12^{4}$ |
| 56.16128.577-56.sp.1.8 | $56$ | $2$ | $2$ | $577$ | $106$ | $1^{60}\cdot2^{52}\cdot4^{10}\cdot6^{8}\cdot12^{4}$ |
| 56.16128.577-56.bjx.1.11 | $56$ | $2$ | $2$ | $577$ | $101$ | $1^{60}\cdot2^{52}\cdot4^{10}\cdot6^{8}\cdot12^{4}$ |
| 56.16128.577-56.bjz.1.10 | $56$ | $2$ | $2$ | $577$ | $110$ | $1^{60}\cdot2^{52}\cdot4^{10}\cdot6^{8}\cdot12^{4}$ |
| 56.16128.577-56.bke.1.9 | $56$ | $2$ | $2$ | $577$ | $101$ | $1^{60}\cdot2^{52}\cdot4^{10}\cdot6^{8}\cdot12^{4}$ |
| 56.16128.577-56.bkf.1.12 | $56$ | $2$ | $2$ | $577$ | $110$ | $1^{60}\cdot2^{52}\cdot4^{10}\cdot6^{8}\cdot12^{4}$ |
| 56.16128.577-56.bph.1.13 | $56$ | $2$ | $2$ | $577$ | $107$ | $1^{76}\cdot2^{50}\cdot4^{13}\cdot6^{8}\cdot12^{2}$ |
| 56.16128.577-56.bpj.1.16 | $56$ | $2$ | $2$ | $577$ | $103$ | $1^{76}\cdot2^{50}\cdot4^{13}\cdot6^{8}\cdot12^{2}$ |
| 56.16128.577-56.bpp.1.15 | $56$ | $2$ | $2$ | $577$ | $107$ | $1^{76}\cdot2^{50}\cdot4^{13}\cdot6^{8}\cdot12^{2}$ |
| 56.16128.577-56.bpr.1.15 | $56$ | $2$ | $2$ | $577$ | $103$ | $1^{76}\cdot2^{50}\cdot4^{13}\cdot6^{8}\cdot12^{2}$ |