Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
| Index: | $2016$ | $\PSL_2$-index: | $2016$ | ||||
| Genus: | $145 = 1 + \frac{ 2016 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 48 }{2}$ | ||||||
| Cusps: | $48$ (none of which are rational) | Cusp widths | $28^{24}\cdot56^{24}$ | Cusp orbits | $6^{4}\cdot12^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $24$ | ||||||
| $\Q$-gonality: | $20 \le \gamma \le 42$ | ||||||
| $\overline{\Q}$-gonality: | $20 \le \gamma \le 42$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.2016.145.861 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}15&48\\30&31\end{bmatrix}$, $\begin{bmatrix}17&32\\16&25\end{bmatrix}$, $\begin{bmatrix}19&20\\16&49\end{bmatrix}$, $\begin{bmatrix}27&4\\6&47\end{bmatrix}$, $\begin{bmatrix}55&0\\0&55\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 56.4032.145-56.bks.2.1, 56.4032.145-56.bks.2.2, 56.4032.145-56.bks.2.3, 56.4032.145-56.bks.2.4, 56.4032.145-56.bks.2.5, 56.4032.145-56.bks.2.6, 56.4032.145-56.bks.2.7, 56.4032.145-56.bks.2.8, 56.4032.145-56.bks.2.9, 56.4032.145-56.bks.2.10, 56.4032.145-56.bks.2.11, 56.4032.145-56.bks.2.12, 56.4032.145-56.bks.2.13, 56.4032.145-56.bks.2.14, 56.4032.145-56.bks.2.15, 56.4032.145-56.bks.2.16 |
| Cyclic 56-isogeny field degree: | $16$ |
| Cyclic 56-torsion field degree: | $192$ |
| Full 56-torsion field degree: | $1536$ |
Jacobian
| Conductor: | $2^{572}\cdot7^{290}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{17}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$ |
| Newforms: | 98.2.a.b$^{4}$, 196.2.a.a, 196.2.a.b$^{2}$, 196.2.a.c$^{3}$, 392.2.a.a, 392.2.a.c, 392.2.a.e, 392.2.a.f, 392.2.a.g$^{2}$, 392.2.b.e$^{2}$, 392.2.b.f, 392.2.b.g$^{3}$, 784.2.a.c, 784.2.a.g, 784.2.a.j, 784.2.a.k, 784.2.a.l, 784.2.a.m, 1568.2.b.e, 1568.2.b.g, 3136.2.a.b, 3136.2.a.bc, 3136.2.a.be, 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.bx, 3136.2.a.bz, 3136.2.a.h, 3136.2.a.j, 3136.2.a.m, 3136.2.a.s, 3136.2.a.u |
Rational points
This modular curve has no $\Q_p$ points for $p=3,11,23,37,67,103,149,193,233$, 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 |
|---|---|---|---|---|---|---|
| 56.96.1.by.1 | $56$ | $21$ | $21$ | $1$ | $1$ | $1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$ |
| 56.1008.70.bd.2 | $56$ | $2$ | $2$ | $70$ | $6$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.1008.70.bf.1 | $56$ | $2$ | $2$ | $70$ | $6$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.1008.70.ey.2 | $56$ | $2$ | $2$ | $70$ | $10$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.1008.70.fd.1 | $56$ | $2$ | $2$ | $70$ | $10$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.1008.73.hd.1 | $56$ | $2$ | $2$ | $73$ | $10$ | $1^{12}\cdot2^{12}\cdot6^{2}\cdot12^{2}$ |
| 56.1008.73.hf.2 | $56$ | $2$ | $2$ | $73$ | $10$ | $1^{12}\cdot2^{12}\cdot6^{2}\cdot12^{2}$ |
| 56.1008.73.jc.1 | $56$ | $2$ | $2$ | $73$ | $24$ | $6^{4}\cdot12^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.4032.289.bdw.2 | $56$ | $2$ | $2$ | $289$ | $53$ | $1^{50}\cdot2^{19}\cdot4^{8}\cdot6^{4}$ |
| 56.4032.289.bea.1 | $56$ | $2$ | $2$ | $289$ | $60$ | $1^{50}\cdot2^{19}\cdot4^{8}\cdot6^{4}$ |
| 56.4032.289.bfc.2 | $56$ | $2$ | $2$ | $289$ | $45$ | $1^{50}\cdot2^{19}\cdot4^{8}\cdot6^{4}$ |
| 56.4032.289.bfg.2 | $56$ | $2$ | $2$ | $289$ | $58$ | $1^{50}\cdot2^{19}\cdot4^{8}\cdot6^{4}$ |
| 56.4032.289.bms.2 | $56$ | $2$ | $2$ | $289$ | $50$ | $1^{50}\cdot2^{19}\cdot4^{8}\cdot6^{4}$ |
| 56.4032.289.bmw.2 | $56$ | $2$ | $2$ | $289$ | $54$ | $1^{50}\cdot2^{19}\cdot4^{8}\cdot6^{4}$ |
| 56.4032.289.bou.2 | $56$ | $2$ | $2$ | $289$ | $52$ | $1^{50}\cdot2^{19}\cdot4^{8}\cdot6^{4}$ |
| 56.4032.289.boy.2 | $56$ | $2$ | $2$ | $289$ | $56$ | $1^{50}\cdot2^{19}\cdot4^{8}\cdot6^{4}$ |