Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
| 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: | $0$ | ||||||
| $\Q$-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: | 56.504.16.37 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}13&4\\18&11\end{bmatrix}$, $\begin{bmatrix}41&32\\14&43\end{bmatrix}$, $\begin{bmatrix}42&1\\43&52\end{bmatrix}$, $\begin{bmatrix}47&54\\26&27\end{bmatrix}$, $\begin{bmatrix}48&51\\23&36\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.252.16.ci.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: | $2^{64}\cdot7^{32}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}\cdot2^{6}$ |
| Newforms: | 98.2.a.b$^{2}$, 196.2.a.b, 196.2.a.c, 3136.2.a.bc, 3136.2.a.bm, 3136.2.a.bp, 3136.2.a.bs, 3136.2.a.j, 3136.2.a.s |
Rational points
This modular curve has no $\Q_p$ points for $p=3,11,67$, 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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(7)$ | $7$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 8.24.0-8.m.1.8 | $8$ | $21$ | $21$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.24.0-8.m.1.8 | $8$ | $21$ | $21$ | $0$ | $0$ | full Jacobian |
| 28.252.7-28.c.1.5 | $28$ | $2$ | $2$ | $7$ | $0$ | $1^{3}\cdot2^{3}$ |
| 56.252.7-28.c.1.19 | $56$ | $2$ | $2$ | $7$ | $0$ | $1^{3}\cdot2^{3}$ |
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.nw.1.16 | $56$ | $2$ | $2$ | $31$ | $10$ | $1^{13}\cdot2$ |
| 56.1008.31-56.ny.1.16 | $56$ | $2$ | $2$ | $31$ | $9$ | $1^{13}\cdot2$ |
| 56.1008.31-56.oa.1.16 | $56$ | $2$ | $2$ | $31$ | $6$ | $1^{13}\cdot2$ |
| 56.1008.31-56.oc.1.16 | $56$ | $2$ | $2$ | $31$ | $3$ | $1^{13}\cdot2$ |
| 56.1008.31-56.pc.1.24 | $56$ | $2$ | $2$ | $31$ | $4$ | $1^{13}\cdot2$ |
| 56.1008.31-56.pe.1.16 | $56$ | $2$ | $2$ | $31$ | $4$ | $1^{13}\cdot2$ |
| 56.1008.31-56.pg.1.16 | $56$ | $2$ | $2$ | $31$ | $7$ | $1^{13}\cdot2$ |
| 56.1008.31-56.pi.1.16 | $56$ | $2$ | $2$ | $31$ | $5$ | $1^{13}\cdot2$ |
| 56.1008.34-56.cg.1.1 | $56$ | $2$ | $2$ | $34$ | $10$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.ci.1.7 | $56$ | $2$ | $2$ | $34$ | $4$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.dg.1.1 | $56$ | $2$ | $2$ | $34$ | $4$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.di.1.7 | $56$ | $2$ | $2$ | $34$ | $10$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.ey.1.8 | $56$ | $2$ | $2$ | $34$ | $8$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.fa.1.7 | $56$ | $2$ | $2$ | $34$ | $6$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.fc.1.7 | $56$ | $2$ | $2$ | $34$ | $6$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.fe.1.7 | $56$ | $2$ | $2$ | $34$ | $8$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.gm.1.16 | $56$ | $2$ | $2$ | $34$ | $5$ | $1^{14}\cdot2^{2}$ |
| 56.1008.34-56.go.1.16 | $56$ | $2$ | $2$ | $34$ | $8$ | $1^{14}\cdot2^{2}$ |
| 56.1008.34-56.gq.1.16 | $56$ | $2$ | $2$ | $34$ | $8$ | $1^{14}\cdot2^{2}$ |
| 56.1008.34-56.gs.1.16 | $56$ | $2$ | $2$ | $34$ | $5$ | $1^{14}\cdot2^{2}$ |
| 56.1008.34-56.hs.1.16 | $56$ | $2$ | $2$ | $34$ | $3$ | $1^{14}\cdot2^{2}$ |
| 56.1008.34-56.hu.1.16 | $56$ | $2$ | $2$ | $34$ | $12$ | $1^{14}\cdot2^{2}$ |
| 56.1008.34-56.hw.1.16 | $56$ | $2$ | $2$ | $34$ | $12$ | $1^{14}\cdot2^{2}$ |
| 56.1008.34-56.hy.1.15 | $56$ | $2$ | $2$ | $34$ | $3$ | $1^{14}\cdot2^{2}$ |