Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $392$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $11 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $4$ are rational) | Cusp widths | $2^{2}\cdot4^{3}\cdot8\cdot14^{2}\cdot28^{3}\cdot56$ | Cusp orbits | $1^{4}\cdot2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 8$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 56R11 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.384.11.2771 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}9&36\\0&39\end{bmatrix}$, $\begin{bmatrix}11&16\\28&27\end{bmatrix}$, $\begin{bmatrix}13&28\\14&1\end{bmatrix}$, $\begin{bmatrix}19&28\\14&3\end{bmatrix}$, $\begin{bmatrix}31&48\\42&39\end{bmatrix}$, $\begin{bmatrix}41&12\\28&47\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.192.11.r.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $2$ |
| Cyclic 56-torsion field degree: | $48$ |
| Full 56-torsion field degree: | $8064$ |
Jacobian
| Conductor: | $2^{27}\cdot7^{17}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}\cdot2\cdot4$ |
| Newforms: | 14.2.a.a$^{3}$, 56.2.a.a, 56.2.a.b, 392.2.b.b, 392.2.b.c |
Models
Canonical model in $\mathbb{P}^{ 10 }$ defined by 36 equations
| $ 0 $ | $=$ | $ x u - x s - x b + y a + z a - w a $ |
| $=$ | $x r + y r - z a + w a - t r + t a$ | |
| $=$ | $x r + y r - y a - z r + t a$ | |
| $=$ | $x y - x z - 2 x u - x r - y a + t a$ | |
| $=$ | $\cdots$ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:0:0:0:0:0:-1:0:1)$, $(0:0:0:0:0:0:0:-1:0:0:1)$, $(0:0:0:0:0:0:0:0:0:0:1)$, $(0:0:0:0:0:0:0:0:0:1:1)$ |
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 28.96.5.b.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
| $\displaystyle W$ | $=$ | $\displaystyle -w$ |
| $\displaystyle T$ | $=$ | $\displaystyle t$ |
Equation of the image curve:
| $0$ | $=$ | $ XY+YZ+YW+ZT $ |
| $=$ | $ Y^{2}-XW-2YT+T^{2} $ | |
| $=$ | $ Y^{2}+XZ+Z^{2}+XW+ZW-YT+T^{2} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.48.0-56.i.1.6 | $56$ | $8$ | $8$ | $0$ | $0$ | full Jacobian |
| 56.192.5-28.b.1.15 | $56$ | $2$ | $2$ | $5$ | $0$ | $2\cdot4$ |
| 56.192.5-28.b.1.25 | $56$ | $2$ | $2$ | $5$ | $0$ | $2\cdot4$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.768.21-56.bb.1.16 | $56$ | $2$ | $2$ | $21$ | $0$ | $2^{3}\cdot4$ |
| 56.768.21-56.bb.3.15 | $56$ | $2$ | $2$ | $21$ | $0$ | $2^{3}\cdot4$ |
| 56.768.21-56.bf.1.16 | $56$ | $2$ | $2$ | $21$ | $0$ | $2^{3}\cdot4$ |
| 56.768.21-56.bf.3.15 | $56$ | $2$ | $2$ | $21$ | $0$ | $2^{3}\cdot4$ |
| 56.768.21-56.bj.2.14 | $56$ | $2$ | $2$ | $21$ | $0$ | $2^{3}\cdot4$ |
| 56.768.21-56.bj.4.13 | $56$ | $2$ | $2$ | $21$ | $0$ | $2^{3}\cdot4$ |
| 56.768.21-56.bn.2.12 | $56$ | $2$ | $2$ | $21$ | $0$ | $2^{3}\cdot4$ |
| 56.768.21-56.bn.4.11 | $56$ | $2$ | $2$ | $21$ | $0$ | $2^{3}\cdot4$ |
| 56.768.23-56.f.2.31 | $56$ | $2$ | $2$ | $23$ | $1$ | $1^{6}\cdot2\cdot4$ |
| 56.768.23-56.h.1.18 | $56$ | $2$ | $2$ | $23$ | $1$ | $1^{6}\cdot2\cdot4$ |
| 56.768.23-56.m.1.27 | $56$ | $2$ | $2$ | $23$ | $2$ | $1^{6}\cdot2\cdot4$ |
| 56.768.23-56.o.1.30 | $56$ | $2$ | $2$ | $23$ | $2$ | $1^{6}\cdot2\cdot4$ |
| 56.768.23-56.v.1.31 | $56$ | $2$ | $2$ | $23$ | $2$ | $1^{6}\cdot2\cdot4$ |
| 56.768.23-56.y.1.31 | $56$ | $2$ | $2$ | $23$ | $2$ | $1^{6}\cdot2\cdot4$ |
| 56.768.23-56.ba.2.27 | $56$ | $2$ | $2$ | $23$ | $3$ | $1^{6}\cdot2\cdot4$ |
| 56.768.23-56.bd.2.30 | $56$ | $2$ | $2$ | $23$ | $3$ | $1^{6}\cdot2\cdot4$ |
| 56.768.23-56.bi.1.16 | $56$ | $2$ | $2$ | $23$ | $0$ | $2^{4}\cdot4$ |
| 56.768.23-56.bi.3.15 | $56$ | $2$ | $2$ | $23$ | $0$ | $2^{4}\cdot4$ |
| 56.768.23-56.bm.1.16 | $56$ | $2$ | $2$ | $23$ | $2$ | $2^{4}\cdot4$ |
| 56.768.23-56.bm.3.15 | $56$ | $2$ | $2$ | $23$ | $2$ | $2^{4}\cdot4$ |
| 56.768.23-56.bq.2.16 | $56$ | $2$ | $2$ | $23$ | $0$ | $2^{4}\cdot4$ |
| 56.768.23-56.bq.4.15 | $56$ | $2$ | $2$ | $23$ | $0$ | $2^{4}\cdot4$ |
| 56.768.23-56.bu.2.16 | $56$ | $2$ | $2$ | $23$ | $2$ | $2^{4}\cdot4$ |
| 56.768.23-56.bu.4.15 | $56$ | $2$ | $2$ | $23$ | $2$ | $2^{4}\cdot4$ |
| 56.768.23-56.by.1.22 | $56$ | $2$ | $2$ | $23$ | $2$ | $2^{4}\cdot4$ |
| 56.768.23-56.by.2.22 | $56$ | $2$ | $2$ | $23$ | $2$ | $2^{4}\cdot4$ |
| 56.768.23-56.cc.1.22 | $56$ | $2$ | $2$ | $23$ | $0$ | $2^{4}\cdot4$ |
| 56.768.23-56.cc.2.22 | $56$ | $2$ | $2$ | $23$ | $0$ | $2^{4}\cdot4$ |
| 56.768.23-56.cg.1.20 | $56$ | $2$ | $2$ | $23$ | $2$ | $2^{4}\cdot4$ |
| 56.768.23-56.cg.2.20 | $56$ | $2$ | $2$ | $23$ | $2$ | $2^{4}\cdot4$ |
| 56.768.23-56.ck.1.22 | $56$ | $2$ | $2$ | $23$ | $0$ | $2^{4}\cdot4$ |
| 56.768.23-56.ck.2.22 | $56$ | $2$ | $2$ | $23$ | $0$ | $2^{4}\cdot4$ |
| 56.768.23-56.cp.2.14 | $56$ | $2$ | $2$ | $23$ | $3$ | $1^{6}\cdot2\cdot4$ |
| 56.768.23-56.cs.2.10 | $56$ | $2$ | $2$ | $23$ | $3$ | $1^{6}\cdot2\cdot4$ |
| 56.768.23-56.cu.2.12 | $56$ | $2$ | $2$ | $23$ | $2$ | $1^{6}\cdot2\cdot4$ |
| 56.768.23-56.cx.1.16 | $56$ | $2$ | $2$ | $23$ | $2$ | $1^{6}\cdot2\cdot4$ |
| 56.768.23-56.df.1.14 | $56$ | $2$ | $2$ | $23$ | $2$ | $1^{6}\cdot2\cdot4$ |
| 56.768.23-56.dh.1.10 | $56$ | $2$ | $2$ | $23$ | $2$ | $1^{6}\cdot2\cdot4$ |
| 56.768.23-56.dm.2.14 | $56$ | $2$ | $2$ | $23$ | $1$ | $1^{6}\cdot2\cdot4$ |
| 56.768.23-56.do.1.16 | $56$ | $2$ | $2$ | $23$ | $1$ | $1^{6}\cdot2\cdot4$ |
| 56.768.25-56.br.2.28 | $56$ | $2$ | $2$ | $25$ | $1$ | $1^{4}\cdot2^{3}\cdot4$ |
| 56.768.25-56.bx.2.20 | $56$ | $2$ | $2$ | $25$ | $1$ | $1^{4}\cdot2^{3}\cdot4$ |
| 56.768.25-56.cj.2.26 | $56$ | $2$ | $2$ | $25$ | $3$ | $1^{4}\cdot2^{3}\cdot4$ |
| 56.768.25-56.cl.1.26 | $56$ | $2$ | $2$ | $25$ | $3$ | $1^{4}\cdot2^{3}\cdot4$ |
| 56.768.25-56.dr.2.30 | $56$ | $2$ | $2$ | $25$ | $4$ | $1^{4}\cdot2^{3}\cdot4$ |
| 56.768.25-56.ea.2.28 | $56$ | $2$ | $2$ | $25$ | $4$ | $1^{4}\cdot2^{3}\cdot4$ |
| 56.768.25-56.ef.1.26 | $56$ | $2$ | $2$ | $25$ | $4$ | $1^{4}\cdot2^{3}\cdot4$ |
| 56.768.25-56.ei.1.26 | $56$ | $2$ | $2$ | $25$ | $4$ | $1^{4}\cdot2^{3}\cdot4$ |
| 56.768.25-56.fc.1.12 | $56$ | $2$ | $2$ | $25$ | $0$ | $2\cdot4\cdot8$ |
| 56.768.25-56.fc.2.12 | $56$ | $2$ | $2$ | $25$ | $0$ | $2\cdot4\cdot8$ |
| 56.768.25-56.fg.1.12 | $56$ | $2$ | $2$ | $25$ | $0$ | $2\cdot4\cdot8$ |
| 56.768.25-56.fg.2.12 | $56$ | $2$ | $2$ | $25$ | $0$ | $2\cdot4\cdot8$ |
| 56.768.25-56.fk.1.8 | $56$ | $2$ | $2$ | $25$ | $0$ | $2\cdot4\cdot8$ |
| 56.768.25-56.fk.2.8 | $56$ | $2$ | $2$ | $25$ | $0$ | $2\cdot4\cdot8$ |
| 56.768.25-56.fo.1.12 | $56$ | $2$ | $2$ | $25$ | $0$ | $2\cdot4\cdot8$ |
| 56.768.25-56.fo.2.12 | $56$ | $2$ | $2$ | $25$ | $0$ | $2\cdot4\cdot8$ |
| 56.1152.31-56.ci.2.23 | $56$ | $3$ | $3$ | $31$ | $0$ | $2^{4}\cdot12$ |
| 56.1152.31-56.ci.4.19 | $56$ | $3$ | $3$ | $31$ | $0$ | $2^{4}\cdot12$ |
| 56.1152.31-56.cp.1.32 | $56$ | $3$ | $3$ | $31$ | $2$ | $1^{8}\cdot6^{2}$ |
| 56.2688.89-56.eh.1.18 | $56$ | $7$ | $7$ | $89$ | $5$ | $1^{22}\cdot2^{10}\cdot4^{3}\cdot6^{2}\cdot12$ |