Invariants
| Level: | $56$ | $\SL_2$-level: | $28$ | Newform level: | $3136$ | ||
| Index: | $1344$ | $\PSL_2$-index: | $672$ | ||||
| Genus: | $45 = 1 + \frac{ 672 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (none of which are rational) | Cusp widths | $28^{24}$ | Cusp orbits | $2^{3}\cdot6^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $10$ | ||||||
| $\Q$-gonality: | $12 \le \gamma \le 24$ | ||||||
| $\overline{\Q}$-gonality: | $12 \le \gamma \le 24$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.1344.45.669 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}1&48\\52&41\end{bmatrix}$, $\begin{bmatrix}11&28\\42&25\end{bmatrix}$, $\begin{bmatrix}31&46\\26&11\end{bmatrix}$, $\begin{bmatrix}33&0\\44&31\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.672.45.j.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $8$ |
| Cyclic 56-torsion field degree: | $96$ |
| Full 56-torsion field degree: | $2304$ |
Jacobian
| Conductor: | $2^{183}\cdot7^{85}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{21}\cdot2^{12}$ |
| Newforms: | 14.2.a.a$^{2}$, 98.2.a.a, 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.b, 392.2.a.d, 392.2.a.e, 392.2.a.g, 448.2.a.a, 448.2.a.e, 448.2.a.h, 3136.2.a.bb, 3136.2.a.bc, 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.e, 3136.2.a.i, 3136.2.a.j, 3136.2.a.q, 3136.2.a.s, 3136.2.a.v, 3136.2.a.w |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=47$, 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 |
|---|---|---|---|---|---|---|
| 28.672.21-28.d.1.6 | $28$ | $2$ | $2$ | $21$ | $3$ | $1^{12}\cdot2^{6}$ |
| 56.48.0-56.g.1.5 | $56$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
| 56.672.21-56.b.1.1 | $56$ | $2$ | $2$ | $21$ | $1$ | $1^{12}\cdot2^{6}$ |
| 56.672.21-56.b.1.4 | $56$ | $2$ | $2$ | $21$ | $1$ | $1^{12}\cdot2^{6}$ |
| 56.672.21-56.c.1.1 | $56$ | $2$ | $2$ | $21$ | $6$ | $1^{12}\cdot2^{6}$ |
| 56.672.21-56.c.1.12 | $56$ | $2$ | $2$ | $21$ | $6$ | $1^{12}\cdot2^{6}$ |
| 56.672.21-28.d.1.8 | $56$ | $2$ | $2$ | $21$ | $3$ | $1^{12}\cdot2^{6}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.2688.89-56.by.1.8 | $56$ | $2$ | $2$ | $89$ | $27$ | $1^{38}\cdot2^{3}$ |
| 56.2688.89-56.cb.1.3 | $56$ | $2$ | $2$ | $89$ | $31$ | $1^{38}\cdot2^{3}$ |
| 56.2688.89-56.ck.1.1 | $56$ | $2$ | $2$ | $89$ | $29$ | $1^{38}\cdot2^{3}$ |
| 56.2688.89-56.cn.1.1 | $56$ | $2$ | $2$ | $89$ | $20$ | $1^{38}\cdot2^{3}$ |
| 56.2688.89-56.dg.1.10 | $56$ | $2$ | $2$ | $89$ | $21$ | $1^{38}\cdot2^{3}$ |
| 56.2688.89-56.dj.1.3 | $56$ | $2$ | $2$ | $89$ | $31$ | $1^{38}\cdot2^{3}$ |
| 56.2688.89-56.ds.1.1 | $56$ | $2$ | $2$ | $89$ | $33$ | $1^{38}\cdot2^{3}$ |
| 56.2688.89-56.dv.1.1 | $56$ | $2$ | $2$ | $89$ | $20$ | $1^{38}\cdot2^{3}$ |
| 56.4032.133-56.cc.1.5 | $56$ | $3$ | $3$ | $133$ | $37$ | $1^{58}\cdot2^{15}$ |