Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
| Index: | $2016$ | $\PSL_2$-index: | $1008$ | ||||
| Genus: | $70 = 1 + \frac{ 1008 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 30 }{2}$ | ||||||
| Cusps: | $30$ (none of which are rational) | Cusp widths | $28^{24}\cdot56^{6}$ | Cusp orbits | $3^{2}\cdot6^{2}\cdot12$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $10$ | ||||||
| $\Q$-gonality: | $10 \le \gamma \le 21$ | ||||||
| $\overline{\Q}$-gonality: | $10 \le \gamma \le 21$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.2016.70.1360 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}15&48\\52&41\end{bmatrix}$, $\begin{bmatrix}17&40\\18&39\end{bmatrix}$, $\begin{bmatrix}23&32\\18&19\end{bmatrix}$, $\begin{bmatrix}33&16\\36&9\end{bmatrix}$, $\begin{bmatrix}49&36\\12&47\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.1008.70.bn.2 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $16$ |
| Cyclic 56-torsion field degree: | $384$ |
| Full 56-torsion field degree: | $1536$ |
Jacobian
| Conductor: | $2^{246}\cdot7^{140}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{10}\cdot2^{12}\cdot6^{2}\cdot12^{2}$ |
| Newforms: | 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, 392.2.b.f, 392.2.b.g$^{2}$, 3136.2.a.a, 3136.2.a.bb, 3136.2.a.bk, 3136.2.a.bm, 3136.2.a.bn, 3136.2.a.bp, 3136.2.a.br, 3136.2.a.bs, 3136.2.a.i, 3136.2.a.k, 3136.2.a.t, 3136.2.a.v |
Rational points
This modular curve has no $\Q_p$ points for $p=3,5,11,23,67$, 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 |
|---|---|---|---|---|---|---|
| 56.96.0-56.n.2.14 | $56$ | $21$ | $21$ | $0$ | $0$ | full Jacobian |
| 56.1008.34-56.h.1.5 | $56$ | $2$ | $2$ | $34$ | $10$ | $6^{2}\cdot12^{2}$ |
| 56.1008.34-56.h.1.35 | $56$ | $2$ | $2$ | $34$ | $10$ | $6^{2}\cdot12^{2}$ |
| 56.1008.34-56.y.2.32 | $56$ | $2$ | $2$ | $34$ | $1$ | $1^{6}\cdot2^{6}\cdot6\cdot12$ |
| 56.1008.34-56.y.2.36 | $56$ | $2$ | $2$ | $34$ | $1$ | $1^{6}\cdot2^{6}\cdot6\cdot12$ |
| 56.1008.34-56.z.2.27 | $56$ | $2$ | $2$ | $34$ | $1$ | $1^{6}\cdot2^{6}\cdot6\cdot12$ |
| 56.1008.34-56.z.2.40 | $56$ | $2$ | $2$ | $34$ | $1$ | $1^{6}\cdot2^{6}\cdot6\cdot12$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.4032.139-56.em.2.15 | $56$ | $2$ | $2$ | $139$ | $22$ | $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.139-56.eq.2.16 | $56$ | $2$ | $2$ | $139$ | $30$ | $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.139-56.fd.1.11 | $56$ | $2$ | $2$ | $139$ | $27$ | $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.139-56.fh.1.12 | $56$ | $2$ | $2$ | $139$ | $15$ | $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.139-56.gb.2.16 | $56$ | $2$ | $2$ | $139$ | $20$ | $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.139-56.gf.2.15 | $56$ | $2$ | $2$ | $139$ | $23$ | $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.139-56.gv.2.11 | $56$ | $2$ | $2$ | $139$ | $26$ | $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.139-56.gz.2.9 | $56$ | $2$ | $2$ | $139$ | $21$ | $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.pb.2.1 | $56$ | $2$ | $2$ | $145$ | $27$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.4032.145-56.pq.2.7 | $56$ | $2$ | $2$ | $145$ | $27$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.4032.145-56.tg.2.5 | $56$ | $2$ | $2$ | $145$ | $23$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.4032.145-56.ti.2.2 | $56$ | $2$ | $2$ | $145$ | $23$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.4032.145-56.bbd.2.12 | $56$ | $2$ | $2$ | $145$ | $27$ | $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.bbf.1.14 | $56$ | $2$ | $2$ | $145$ | $27$ | $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.bbt.1.16 | $56$ | $2$ | $2$ | $145$ | $26$ | $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.bbv.2.11 | $56$ | $2$ | $2$ | $145$ | $26$ | $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.biv.2.10 | $56$ | $2$ | $2$ | $145$ | $26$ | $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.bix.1.10 | $56$ | $2$ | $2$ | $145$ | $26$ | $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.bjl.1.14 | $56$ | $2$ | $2$ | $145$ | $27$ | $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.bjn.2.9 | $56$ | $2$ | $2$ | $145$ | $27$ | $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.bkr.2.2 | $56$ | $2$ | $2$ | $145$ | $28$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.4032.145-56.bkt.2.8 | $56$ | $2$ | $2$ | $145$ | $28$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.4032.145-56.bli.2.7 | $56$ | $2$ | $2$ | $145$ | $24$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.4032.145-56.blj.2.4 | $56$ | $2$ | $2$ | $145$ | $24$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |