Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $1568$ | ||
| Index: | $2688$ | $\PSL_2$-index: | $1344$ | ||||
| Genus: | $93 = 1 + \frac{ 1344 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 40 }{2}$ | ||||||
| Cusps: | $40$ (none of which are rational) | Cusp widths | $14^{16}\cdot28^{8}\cdot56^{16}$ | Cusp orbits | $2^{5}\cdot6^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $7$ | ||||||
| $\Q$-gonality: | $14 \le \gamma \le 48$ | ||||||
| $\overline{\Q}$-gonality: | $14 \le \gamma \le 28$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.2688.93.59 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}11&22\\34&31\end{bmatrix}$, $\begin{bmatrix}27&30\\2&15\end{bmatrix}$, $\begin{bmatrix}45&50\\0&15\end{bmatrix}$, $\begin{bmatrix}53&24\\20&17\end{bmatrix}$, $\begin{bmatrix}53&26\\48&31\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.1344.93.fe.2 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $2$ |
| Cyclic 56-torsion field degree: | $48$ |
| Full 56-torsion field degree: | $1152$ |
Jacobian
| Conductor: | $2^{354}\cdot7^{163}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{21}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| Newforms: | 14.2.a.a$^{4}$, 56.2.a.a$^{2}$, 56.2.a.b$^{2}$, 98.2.a.b$^{4}$, 112.2.a.a, 112.2.a.b, 112.2.a.c, 196.2.a.b$^{3}$, 196.2.a.c$^{3}$, 224.2.b.a$^{2}$, 224.2.b.b$^{2}$, 392.2.a.c$^{2}$, 392.2.a.f$^{2}$, 392.2.a.g$^{2}$, 784.2.a.a, 784.2.a.d, 784.2.a.h, 784.2.a.k, 784.2.a.l, 784.2.a.m, 1568.2.b.f$^{2}$, 1568.2.b.g$^{2}$ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=31,79,223$, 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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(7)$ | $7$ | $96$ | $48$ | $0$ | $0$ | full Jacobian |
| 8.96.0-8.j.2.2 | $8$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.0-8.j.2.2 | $8$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
| 56.1344.45-56.u.2.2 | $56$ | $2$ | $2$ | $45$ | $1$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
| 56.1344.45-56.u.2.20 | $56$ | $2$ | $2$ | $45$ | $1$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
| 56.1344.45-56.cj.1.3 | $56$ | $2$ | $2$ | $45$ | $7$ | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.1344.45-56.cj.1.6 | $56$ | $2$ | $2$ | $45$ | $7$ | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.1344.45-56.gq.1.16 | $56$ | $2$ | $2$ | $45$ | $1$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
| 56.1344.45-56.gq.1.17 | $56$ | $2$ | $2$ | $45$ | $1$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.5376.185-56.nj.2.13 | $56$ | $2$ | $2$ | $185$ | $18$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.185-56.nq.1.9 | $56$ | $2$ | $2$ | $185$ | $34$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.185-56.ok.2.2 | $56$ | $2$ | $2$ | $185$ | $24$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.185-56.oq.2.2 | $56$ | $2$ | $2$ | $185$ | $19$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.185-56.pl.2.13 | $56$ | $2$ | $2$ | $185$ | $16$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.185-56.ps.1.9 | $56$ | $2$ | $2$ | $185$ | $30$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.185-56.qm.2.3 | $56$ | $2$ | $2$ | $185$ | $24$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.185-56.qs.2.5 | $56$ | $2$ | $2$ | $185$ | $23$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.lj.1.4 | $56$ | $2$ | $2$ | $193$ | $22$ | $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.5376.193-56.ue.1.1 | $56$ | $2$ | $2$ | $193$ | $29$ | $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.5376.193-56.blw.2.2 | $56$ | $2$ | $2$ | $193$ | $26$ | $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.5376.193-56.bme.2.1 | $56$ | $2$ | $2$ | $193$ | $31$ | $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.5376.193-56.btw.2.2 | $56$ | $2$ | $2$ | $193$ | $32$ | $1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.bue.2.2 | $56$ | $2$ | $2$ | $193$ | $24$ | $1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.bvc.2.3 | $56$ | $2$ | $2$ | $193$ | $32$ | $1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.bvk.2.3 | $56$ | $2$ | $2$ | $193$ | $24$ | $1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$ |
| 56.8064.277-56.if.2.2 | $56$ | $3$ | $3$ | $277$ | $28$ | $1^{58}\cdot2^{21}\cdot4^{6}\cdot6^{6}\cdot12^{2}$ |