Invariants
| Level: | $56$ | $\SL_2$-level: | $28$ | Newform level: | $3136$ | ||
| Index: | $672$ | $\PSL_2$-index: | $336$ | ||||
| Genus: | $21 = 1 + \frac{ 336 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $14^{8}\cdot28^{8}$ | Cusp orbits | $2^{2}\cdot6^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $6$ | ||||||
| $\Q$-gonality: | $6 \le \gamma \le 12$ | ||||||
| $\overline{\Q}$-gonality: | $6 \le \gamma \le 12$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 28D21 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.672.21.433 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}2&47\\27&12\end{bmatrix}$, $\begin{bmatrix}10&15\\51&46\end{bmatrix}$, $\begin{bmatrix}27&2\\44&15\end{bmatrix}$, $\begin{bmatrix}38&29\\9&32\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.336.21.y.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $8$ |
| Cyclic 56-torsion field degree: | $192$ |
| Full 56-torsion field degree: | $4608$ |
Jacobian
| Conductor: | $2^{99}\cdot7^{37}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{9}\cdot2^{6}$ |
| Newforms: | 14.2.a.a, 98.2.a.b, 112.2.a.c, 448.2.a.e, 448.2.a.g, 448.2.a.h, 784.2.a.d, 784.2.a.l, 784.2.a.m, 3136.2.a.bc, 3136.2.a.bn, 3136.2.a.bp, 3136.2.a.br, 3136.2.a.h, 3136.2.a.j |
Rational points
This modular curve has no $\Q_p$ points for $p=3$, 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 |
|---|---|---|---|---|---|---|
| 28.336.9-28.a.1.3 | $28$ | $2$ | $2$ | $9$ | $0$ | $1^{6}\cdot2^{3}$ |
| 56.24.0-8.g.1.2 | $56$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
| 56.336.9-28.a.1.6 | $56$ | $2$ | $2$ | $9$ | $0$ | $1^{6}\cdot2^{3}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.1344.41-56.ej.1.4 | $56$ | $2$ | $2$ | $41$ | $16$ | $1^{18}\cdot2$ |
| 56.1344.41-56.ek.1.2 | $56$ | $2$ | $2$ | $41$ | $14$ | $1^{18}\cdot2$ |
| 56.1344.41-56.eq.1.4 | $56$ | $2$ | $2$ | $41$ | $14$ | $1^{18}\cdot2$ |
| 56.1344.41-56.er.1.3 | $56$ | $2$ | $2$ | $41$ | $14$ | $1^{18}\cdot2$ |
| 56.1344.41-56.ex.1.1 | $56$ | $2$ | $2$ | $41$ | $11$ | $1^{18}\cdot2$ |
| 56.1344.41-56.ey.1.1 | $56$ | $2$ | $2$ | $41$ | $14$ | $1^{18}\cdot2$ |
| 56.1344.41-56.fe.1.1 | $56$ | $2$ | $2$ | $41$ | $11$ | $1^{18}\cdot2$ |
| 56.1344.41-56.ff.1.4 | $56$ | $2$ | $2$ | $41$ | $14$ | $1^{18}\cdot2$ |
| 56.2016.61-56.ct.1.6 | $56$ | $3$ | $3$ | $61$ | $22$ | $1^{26}\cdot2^{7}$ |