Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $392$ | ||
| Index: | $504$ | $\PSL_2$-index: | $252$ | ||||
| Genus: | $16 = 1 + \frac{ 252 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $7^{6}\cdot14^{3}\cdot56^{3}$ | Cusp orbits | $3^{2}\cdot6$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $5 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $5 \le \gamma \le 8$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 56B16 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.504.16.122 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}12&25\\23&2\end{bmatrix}$, $\begin{bmatrix}16&33\\29&16\end{bmatrix}$, $\begin{bmatrix}21&52\\16&49\end{bmatrix}$, $\begin{bmatrix}34&19\\29&48\end{bmatrix}$, $\begin{bmatrix}39&34\\24&45\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.252.16.cl.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $16$ |
| Cyclic 56-torsion field degree: | $192$ |
| Full 56-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{30}\cdot7^{32}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}\cdot2^{6}$ |
| Newforms: | 98.2.a.b$^{3}$, 196.2.a.a, 196.2.a.b, 196.2.a.c$^{2}$, 392.2.a.a, 392.2.a.e, 392.2.a.g |
Rational points
This modular curve has no $\Q_p$ points for $p=11,67$, 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 |
|---|---|---|---|---|---|---|
| 28.252.7-28.c.1.5 | $28$ | $2$ | $2$ | $7$ | $0$ | $1^{3}\cdot2^{3}$ |
| 56.24.0-56.z.1.13 | $56$ | $21$ | $21$ | $0$ | $0$ | full Jacobian |
| 56.252.7-28.c.1.3 | $56$ | $2$ | $2$ | $7$ | $0$ | $1^{3}\cdot2^{3}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.1008.31-56.of.1.16 | $56$ | $2$ | $2$ | $31$ | $4$ | $1^{13}\cdot2$ |
| 56.1008.31-56.oh.1.16 | $56$ | $2$ | $2$ | $31$ | $12$ | $1^{13}\cdot2$ |
| 56.1008.31-56.oj.1.16 | $56$ | $2$ | $2$ | $31$ | $12$ | $1^{13}\cdot2$ |
| 56.1008.31-56.ol.1.16 | $56$ | $2$ | $2$ | $31$ | $6$ | $1^{13}\cdot2$ |
| 56.1008.31-56.pl.1.12 | $56$ | $2$ | $2$ | $31$ | $8$ | $1^{13}\cdot2$ |
| 56.1008.31-56.pn.1.12 | $56$ | $2$ | $2$ | $31$ | $3$ | $1^{13}\cdot2$ |
| 56.1008.31-56.pp.1.24 | $56$ | $2$ | $2$ | $31$ | $5$ | $1^{13}\cdot2$ |
| 56.1008.31-56.pr.1.12 | $56$ | $2$ | $2$ | $31$ | $14$ | $1^{13}\cdot2$ |
| 56.1008.34-56.cd.1.2 | $56$ | $2$ | $2$ | $34$ | $6$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.co.1.7 | $56$ | $2$ | $2$ | $34$ | $8$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.dl.1.1 | $56$ | $2$ | $2$ | $34$ | $8$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.dm.1.7 | $56$ | $2$ | $2$ | $34$ | $14$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.fd.1.4 | $56$ | $2$ | $2$ | $34$ | $6$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.fe.1.1 | $56$ | $2$ | $2$ | $34$ | $8$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.fo.1.2 | $56$ | $2$ | $2$ | $34$ | $8$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.fr.1.1 | $56$ | $2$ | $2$ | $34$ | $14$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.gv.1.16 | $56$ | $2$ | $2$ | $34$ | $6$ | $1^{14}\cdot2^{2}$ |
| 56.1008.34-56.gx.1.20 | $56$ | $2$ | $2$ | $34$ | $6$ | $1^{14}\cdot2^{2}$ |
| 56.1008.34-56.gz.1.15 | $56$ | $2$ | $2$ | $34$ | $15$ | $1^{14}\cdot2^{2}$ |
| 56.1008.34-56.hb.1.15 | $56$ | $2$ | $2$ | $34$ | $9$ | $1^{14}\cdot2^{2}$ |
| 56.1008.34-56.ib.1.16 | $56$ | $2$ | $2$ | $34$ | $6$ | $1^{14}\cdot2^{2}$ |
| 56.1008.34-56.id.1.16 | $56$ | $2$ | $2$ | $34$ | $6$ | $1^{14}\cdot2^{2}$ |
| 56.1008.34-56.if.1.15 | $56$ | $2$ | $2$ | $34$ | $15$ | $1^{14}\cdot2^{2}$ |
| 56.1008.34-56.ih.1.16 | $56$ | $2$ | $2$ | $34$ | $9$ | $1^{14}\cdot2^{2}$ |