Invariants
| Level: | $56$ | $\SL_2$-level: | $28$ | Newform level: | $3136$ | ||
| Index: | $1008$ | $\PSL_2$-index: | $504$ | ||||
| Genus: | $34 = 1 + \frac{ 504 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 18 }{2}$ | ||||||
| Cusps: | $18$ (none of which are rational) | Cusp widths | $28^{18}$ | Cusp orbits | $6^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $4$ | ||||||
| $\Q$-gonality: | $9 \le \gamma \le 16$ | ||||||
| $\overline{\Q}$-gonality: | $9 \le \gamma \le 16$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.1008.34.20 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}1&20\\22&43\end{bmatrix}$, $\begin{bmatrix}3&38\\32&31\end{bmatrix}$, $\begin{bmatrix}27&42\\14&1\end{bmatrix}$, $\begin{bmatrix}33&28\\14&51\end{bmatrix}$, $\begin{bmatrix}47&38\\4&19\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.504.34.b.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $32$ |
| Cyclic 56-torsion field degree: | $384$ |
| Full 56-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{154}\cdot7^{68}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{10}\cdot2^{12}$ |
| Newforms: | 98.2.a.b$^{2}$, 196.2.a.b, 196.2.a.c, 784.2.a.a, 784.2.a.d, 784.2.a.h, 784.2.a.k, 784.2.a.l, 784.2.a.m, 3136.2.a.bc$^{2}$, 3136.2.a.bm$^{2}$, 3136.2.a.bp$^{2}$, 3136.2.a.bs$^{2}$, 3136.2.a.j$^{2}$, 3136.2.a.s$^{2}$ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=3,11,23,31,67$, and therefore no rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(7)$ | $7$ | $48$ | $24$ | $0$ | $0$ | full Jacobian |
| 8.48.0-8.b.1.3 | $8$ | $21$ | $21$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.0-8.b.1.3 | $8$ | $21$ | $21$ | $0$ | $0$ | full Jacobian |
| 28.504.16-28.a.1.8 | $28$ | $2$ | $2$ | $16$ | $4$ | $1^{6}\cdot2^{6}$ |
| 56.504.16-28.a.1.4 | $56$ | $2$ | $2$ | $16$ | $4$ | $1^{6}\cdot2^{6}$ |
| 56.504.16-56.b.1.2 | $56$ | $2$ | $2$ | $16$ | $0$ | $1^{6}\cdot2^{6}$ |
| 56.504.16-56.b.1.6 | $56$ | $2$ | $2$ | $16$ | $0$ | $1^{6}\cdot2^{6}$ |
| 56.504.16-56.b.1.15 | $56$ | $2$ | $2$ | $16$ | $0$ | $1^{6}\cdot2^{6}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.2016.67-56.n.1.4 | $56$ | $2$ | $2$ | $67$ | $26$ | $1^{27}\cdot2^{3}$ |
| 56.2016.67-56.p.1.3 | $56$ | $2$ | $2$ | $67$ | $16$ | $1^{27}\cdot2^{3}$ |
| 56.2016.67-56.s.1.4 | $56$ | $2$ | $2$ | $67$ | $16$ | $1^{27}\cdot2^{3}$ |
| 56.2016.67-56.u.1.4 | $56$ | $2$ | $2$ | $67$ | $13$ | $1^{27}\cdot2^{3}$ |
| 56.2016.67-56.x.1.3 | $56$ | $2$ | $2$ | $67$ | $16$ | $1^{27}\cdot2^{3}$ |
| 56.2016.67-56.z.1.4 | $56$ | $2$ | $2$ | $67$ | $17$ | $1^{27}\cdot2^{3}$ |
| 56.2016.67-56.bc.1.3 | $56$ | $2$ | $2$ | $67$ | $13$ | $1^{27}\cdot2^{3}$ |
| 56.2016.67-56.be.1.3 | $56$ | $2$ | $2$ | $67$ | $19$ | $1^{27}\cdot2^{3}$ |
| 56.2016.73-56.cn.1.3 | $56$ | $2$ | $2$ | $73$ | $21$ | $1^{7}\cdot2^{14}\cdot4$ |
| 56.2016.73-56.do.1.4 | $56$ | $2$ | $2$ | $73$ | $22$ | $1^{7}\cdot2^{14}\cdot4$ |
| 56.2016.73-56.dr.1.4 | $56$ | $2$ | $2$ | $73$ | $21$ | $1^{23}\cdot2^{8}$ |
| 56.2016.73-56.du.1.3 | $56$ | $2$ | $2$ | $73$ | $20$ | $1^{23}\cdot2^{8}$ |
| 56.2016.73-56.ej.1.4 | $56$ | $2$ | $2$ | $73$ | $19$ | $1^{23}\cdot2^{8}$ |
| 56.2016.73-56.em.1.3 | $56$ | $2$ | $2$ | $73$ | $22$ | $1^{23}\cdot2^{8}$ |
| 56.2016.73-56.ep.1.3 | $56$ | $2$ | $2$ | $73$ | $18$ | $1^{7}\cdot2^{14}\cdot4$ |
| 56.2016.73-56.es.1.4 | $56$ | $2$ | $2$ | $73$ | $17$ | $1^{7}\cdot2^{14}\cdot4$ |