Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
| Index: | $2016$ | $\PSL_2$-index: | $1008$ | ||||
| Genus: | $73 = 1 + \frac{ 1008 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (none of which are rational) | Cusp widths | $28^{12}\cdot56^{12}$ | Cusp orbits | $3^{4}\cdot6^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $9$ | ||||||
| $\Q$-gonality: | $10 \le \gamma \le 32$ | ||||||
| $\overline{\Q}$-gonality: | $10 \le \gamma \le 32$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.2016.73.142 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}15&2\\2&35\end{bmatrix}$, $\begin{bmatrix}27&14\\14&55\end{bmatrix}$, $\begin{bmatrix}29&34\\4&27\end{bmatrix}$, $\begin{bmatrix}43&12\\54&37\end{bmatrix}$, $\begin{bmatrix}45&48\\38&39\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.1008.73.ei.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $16$ |
| Cyclic 56-torsion field degree: | $192$ |
| Full 56-torsion field degree: | $1536$ |
Jacobian
| Conductor: | $2^{264}\cdot7^{144}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| Newforms: | 64.2.a.a, 98.2.a.b$^{3}$, 196.2.a.b$^{2}$, 196.2.a.c$^{2}$, 392.2.a.c, 392.2.a.f, 392.2.a.g, 392.2.b.e$^{2}$, 392.2.b.g$^{2}$, 3136.2.a.bd, 3136.2.a.bh, 3136.2.a.bi, 3136.2.a.bj, 3136.2.a.bl, 3136.2.a.bo, 3136.2.a.bv, 3136.2.a.bw, 3136.2.a.bz |
Rational points
This modular curve has no $\Q_p$ points for $p=3,5,11,23,37,67,103,149$, 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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(7)$ | $7$ | $96$ | $48$ | $0$ | $0$ | full Jacobian |
| 8.96.1-8.n.1.3 | $8$ | $21$ | $21$ | $1$ | $0$ | $1^{4}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.1-8.n.1.3 | $8$ | $21$ | $21$ | $1$ | $0$ | $1^{4}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.1008.34-56.z.1.3 | $56$ | $2$ | $2$ | $34$ | $1$ | $1\cdot2^{8}\cdot4\cdot6\cdot12$ |
| 56.1008.34-56.z.1.26 | $56$ | $2$ | $2$ | $34$ | $1$ | $1\cdot2^{8}\cdot4\cdot6\cdot12$ |
| 56.1008.34-56.z.2.5 | $56$ | $2$ | $2$ | $34$ | $1$ | $1\cdot2^{8}\cdot4\cdot6\cdot12$ |
| 56.1008.34-56.z.2.27 | $56$ | $2$ | $2$ | $34$ | $1$ | $1\cdot2^{8}\cdot4\cdot6\cdot12$ |
| 56.1008.37-56.d.1.7 | $56$ | $2$ | $2$ | $37$ | $9$ | $6^{2}\cdot12^{2}$ |
| 56.1008.37-56.d.1.8 | $56$ | $2$ | $2$ | $37$ | $9$ | $6^{2}\cdot12^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.4032.145-56.gt.1.4 | $56$ | $2$ | $2$ | $145$ | $24$ | $1^{22}\cdot2^{11}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.he.1.1 | $56$ | $2$ | $2$ | $145$ | $33$ | $1^{22}\cdot2^{11}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.jk.1.8 | $56$ | $2$ | $2$ | $145$ | $21$ | $1^{22}\cdot2^{11}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.jy.1.4 | $56$ | $2$ | $2$ | $145$ | $23$ | $1^{22}\cdot2^{11}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.pd.1.4 | $56$ | $2$ | $2$ | $145$ | $23$ | $1^{12}\cdot2^{12}\cdot6^{2}\cdot12^{2}$ |
| 56.4032.145-56.pd.2.6 | $56$ | $2$ | $2$ | $145$ | $23$ | $1^{12}\cdot2^{12}\cdot6^{2}\cdot12^{2}$ |
| 56.4032.145-56.qg.1.6 | $56$ | $2$ | $2$ | $145$ | $23$ | $1^{12}\cdot2^{12}\cdot6^{2}\cdot12^{2}$ |
| 56.4032.145-56.qg.2.7 | $56$ | $2$ | $2$ | $145$ | $23$ | $1^{12}\cdot2^{12}\cdot6^{2}\cdot12^{2}$ |
| 56.4032.145-56.qw.1.7 | $56$ | $2$ | $2$ | $145$ | $24$ | $1^{28}\cdot2^{8}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.qw.2.6 | $56$ | $2$ | $2$ | $145$ | $24$ | $1^{28}\cdot2^{8}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.rm.1.5 | $56$ | $2$ | $2$ | $145$ | $22$ | $1^{28}\cdot2^{8}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.rm.2.2 | $56$ | $2$ | $2$ | $145$ | $22$ | $1^{28}\cdot2^{8}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.sc.1.7 | $56$ | $2$ | $2$ | $145$ | $22$ | $1^{28}\cdot2^{8}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.sc.2.6 | $56$ | $2$ | $2$ | $145$ | $22$ | $1^{28}\cdot2^{8}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.ss.1.6 | $56$ | $2$ | $2$ | $145$ | $24$ | $1^{28}\cdot2^{8}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.ss.2.4 | $56$ | $2$ | $2$ | $145$ | $24$ | $1^{28}\cdot2^{8}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.ti.1.3 | $56$ | $2$ | $2$ | $145$ | $23$ | $1^{12}\cdot2^{12}\cdot6^{2}\cdot12^{2}$ |
| 56.4032.145-56.ti.2.2 | $56$ | $2$ | $2$ | $145$ | $23$ | $1^{12}\cdot2^{12}\cdot6^{2}\cdot12^{2}$ |
| 56.4032.145-56.tr.1.6 | $56$ | $2$ | $2$ | $145$ | $23$ | $1^{12}\cdot2^{12}\cdot6^{2}\cdot12^{2}$ |
| 56.4032.145-56.tr.2.6 | $56$ | $2$ | $2$ | $145$ | $23$ | $1^{12}\cdot2^{12}\cdot6^{2}\cdot12^{2}$ |
| 56.4032.145-56.vv.1.6 | $56$ | $2$ | $2$ | $145$ | $27$ | $1^{22}\cdot2^{11}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.wg.1.6 | $56$ | $2$ | $2$ | $145$ | $21$ | $1^{22}\cdot2^{11}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.xz.1.7 | $56$ | $2$ | $2$ | $145$ | $29$ | $1^{22}\cdot2^{11}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.yk.1.7 | $56$ | $2$ | $2$ | $145$ | $18$ | $1^{22}\cdot2^{11}\cdot4^{4}\cdot6^{2}$ |