Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
| Index: | $4032$ | $\PSL_2$-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: | 56.4032.145.1535 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}5&36\\20&9\end{bmatrix}$, $\begin{bmatrix}15&32\\6&55\end{bmatrix}$, $\begin{bmatrix}25&16\\52&7\end{bmatrix}$, $\begin{bmatrix}29&36\\12&55\end{bmatrix}$ |
| $\GL_2(\Z/56\Z)$-subgroup: | Group 768.26500 |
| Contains $-I$: | no $\quad$ (see 56.2016.145.bks.2 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $16$ |
| Cyclic 56-torsion field degree: | $192$ |
| Full 56-torsion field degree: | $768$ |
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 |
|---|---|---|---|---|---|---|
| 56.192.1-56.by.1.8 | $56$ | $21$ | $21$ | $1$ | $1$ | $1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$ |
| 56.2016.70-56.bd.2.8 | $56$ | $2$ | $2$ | $70$ | $6$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.2016.70-56.bd.2.20 | $56$ | $2$ | $2$ | $70$ | $6$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.2016.70-56.bf.1.3 | $56$ | $2$ | $2$ | $70$ | $6$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.2016.70-56.bf.1.31 | $56$ | $2$ | $2$ | $70$ | $6$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.2016.70-56.ey.2.2 | $56$ | $2$ | $2$ | $70$ | $10$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.2016.70-56.ey.2.28 | $56$ | $2$ | $2$ | $70$ | $10$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.2016.70-56.fd.1.2 | $56$ | $2$ | $2$ | $70$ | $10$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.2016.70-56.fd.1.32 | $56$ | $2$ | $2$ | $70$ | $10$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.2016.73-56.hd.1.2 | $56$ | $2$ | $2$ | $73$ | $10$ | $1^{12}\cdot2^{12}\cdot6^{2}\cdot12^{2}$ |
| 56.2016.73-56.hd.1.32 | $56$ | $2$ | $2$ | $73$ | $10$ | $1^{12}\cdot2^{12}\cdot6^{2}\cdot12^{2}$ |
| 56.2016.73-56.hf.2.3 | $56$ | $2$ | $2$ | $73$ | $10$ | $1^{12}\cdot2^{12}\cdot6^{2}\cdot12^{2}$ |
| 56.2016.73-56.hf.2.28 | $56$ | $2$ | $2$ | $73$ | $10$ | $1^{12}\cdot2^{12}\cdot6^{2}\cdot12^{2}$ |
| 56.2016.73-56.jc.1.5 | $56$ | $2$ | $2$ | $73$ | $24$ | $6^{4}\cdot12^{4}$ |
| 56.2016.73-56.jc.1.9 | $56$ | $2$ | $2$ | $73$ | $24$ | $6^{4}\cdot12^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.8064.289-56.bdw.2.5 | $56$ | $2$ | $2$ | $289$ | $53$ | $1^{50}\cdot2^{19}\cdot4^{8}\cdot6^{4}$ |
| 56.8064.289-56.bea.1.3 | $56$ | $2$ | $2$ | $289$ | $60$ | $1^{50}\cdot2^{19}\cdot4^{8}\cdot6^{4}$ |
| 56.8064.289-56.bfc.2.6 | $56$ | $2$ | $2$ | $289$ | $45$ | $1^{50}\cdot2^{19}\cdot4^{8}\cdot6^{4}$ |
| 56.8064.289-56.bfg.2.7 | $56$ | $2$ | $2$ | $289$ | $58$ | $1^{50}\cdot2^{19}\cdot4^{8}\cdot6^{4}$ |
| 56.8064.289-56.bms.2.4 | $56$ | $2$ | $2$ | $289$ | $50$ | $1^{50}\cdot2^{19}\cdot4^{8}\cdot6^{4}$ |
| 56.8064.289-56.bmw.2.2 | $56$ | $2$ | $2$ | $289$ | $54$ | $1^{50}\cdot2^{19}\cdot4^{8}\cdot6^{4}$ |
| 56.8064.289-56.bou.2.3 | $56$ | $2$ | $2$ | $289$ | $52$ | $1^{50}\cdot2^{19}\cdot4^{8}\cdot6^{4}$ |
| 56.8064.289-56.boy.2.2 | $56$ | $2$ | $2$ | $289$ | $56$ | $1^{50}\cdot2^{19}\cdot4^{8}\cdot6^{4}$ |