Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
| 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 | $4^{6}\cdot28^{6}$ | Cusp orbits | $1^{4}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $4$ | ||||||
| $\Q$-gonality: | $5 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $5 \le \gamma \le 8$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 28E11 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.384.11.3003 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}5&7\\40&39\end{bmatrix}$, $\begin{bmatrix}21&44\\32&33\end{bmatrix}$, $\begin{bmatrix}27&13\\36&5\end{bmatrix}$, $\begin{bmatrix}37&17\\16&39\end{bmatrix}$, $\begin{bmatrix}41&34\\32&45\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.192.11.dc.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $2$ |
| Cyclic 56-torsion field degree: | $24$ |
| Full 56-torsion field degree: | $8064$ |
Jacobian
| Conductor: | $2^{50}\cdot7^{17}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{11}$ |
| Newforms: | 14.2.a.a$^{2}$, 112.2.a.a, 112.2.a.b, 112.2.a.c, 3136.2.a.c, 3136.2.a.e, 3136.2.a.p, 3136.2.a.q, 3136.2.a.w, 3136.2.a.z |
Models
Canonical model in $\mathbb{P}^{ 10 }$ defined by 36 equations
| $ 0 $ | $=$ | $ u r + s a $ |
| $=$ | $y r + z a$ | |
| $=$ | $x u - z s$ | |
| $=$ | $x a + z r$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 16 x^{13} z + 224 x^{12} y^{2} + 128 x^{11} z^{3} + 5600 x^{10} y^{2} z^{2} - 2401 x^{9} y^{4} z + \cdots + 889056 y^{6} z^{8} $ |
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 |
|---|
| $(0:0:0:0:0:0:0:0:1:0:0)$, $(0:0:0:0:0:0:0:0:0:-1:1)$, $(0:0:0:0:0:1:1:0:0:0:0)$, $(0:0:0:0:0:0:0:1:0:0:0)$ |
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.h.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y$ |
| $\displaystyle W$ | $=$ | $\displaystyle z-t$ |
| $\displaystyle T$ | $=$ | $\displaystyle -w$ |
Equation of the image curve:
| $0$ | $=$ | $ Y^{2}-XZ $ |
| $=$ | $ XY-ZW-YT+WT $ | |
| $=$ | $ X^{2}+3Y^{2}+3XZ-2YW+W^{2}+2ZT-T^{2} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 56.192.11.dc.1 :
| $\displaystyle X$ | $=$ | $\displaystyle a+b$ |
| $\displaystyle Y$ | $=$ | $\displaystyle t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle r$ |
Equation of the image curve:
| $0$ | $=$ | $ 224X^{12}Y^{2}-10290X^{8}Y^{6}-504210X^{4}Y^{10}+26353376Y^{14}-16X^{13}Z-2401X^{9}Y^{4}Z-674681X^{5}Y^{8}Z+30118144XY^{12}Z+5600X^{10}Y^{2}Z^{2}-329280X^{6}Y^{6}Z^{2}+18420472X^{2}Y^{10}Z^{2}+128X^{11}Z^{3}-161798X^{7}Y^{4}Z^{3}+5512696X^{3}Y^{8}Z^{3}+2072X^{8}Y^{2}Z^{4}+1299970X^{4}Y^{6}Z^{4}+4840416Y^{10}Z^{4}-328X^{9}Z^{5}-131663X^{5}Y^{4}Z^{5}+2074464XY^{8}Z^{5}+1484X^{6}Y^{2}Z^{6}+1506456X^{2}Y^{6}Z^{6}+288X^{7}Z^{7}-31752X^{3}Y^{4}Z^{7}-10458X^{4}Y^{2}Z^{8}+889056Y^{6}Z^{8}-81X^{5}Z^{9}+127008XY^{4}Z^{9}+4536X^{2}Y^{2}Z^{10} $ |
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.x.1.8 | $56$ | $8$ | $8$ | $0$ | $0$ | full Jacobian |
| 56.192.5-28.h.1.11 | $56$ | $2$ | $2$ | $5$ | $1$ | $1^{6}$ |
| 56.192.5-28.h.1.27 | $56$ | $2$ | $2$ | $5$ | $1$ | $1^{6}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.768.21-56.dd.1.6 | $56$ | $2$ | $2$ | $21$ | $6$ | $2^{5}$ |
| 56.768.21-56.dd.1.15 | $56$ | $2$ | $2$ | $21$ | $6$ | $2^{5}$ |
| 56.768.21-56.dd.2.6 | $56$ | $2$ | $2$ | $21$ | $6$ | $2^{5}$ |
| 56.768.21-56.dd.2.15 | $56$ | $2$ | $2$ | $21$ | $6$ | $2^{5}$ |
| 56.768.21-56.de.1.8 | $56$ | $2$ | $2$ | $21$ | $4$ | $2^{5}$ |
| 56.768.21-56.de.1.13 | $56$ | $2$ | $2$ | $21$ | $4$ | $2^{5}$ |
| 56.768.21-56.de.2.8 | $56$ | $2$ | $2$ | $21$ | $4$ | $2^{5}$ |
| 56.768.21-56.de.2.13 | $56$ | $2$ | $2$ | $21$ | $4$ | $2^{5}$ |
| 56.768.25-56.ht.1.6 | $56$ | $2$ | $2$ | $25$ | $10$ | $1^{10}\cdot2^{2}$ |
| 56.768.25-56.hz.1.9 | $56$ | $2$ | $2$ | $25$ | $8$ | $1^{10}\cdot2^{2}$ |
| 56.768.25-56.jb.1.2 | $56$ | $2$ | $2$ | $25$ | $11$ | $1^{10}\cdot2^{2}$ |
| 56.768.25-56.ji.1.1 | $56$ | $2$ | $2$ | $25$ | $9$ | $1^{10}\cdot2^{2}$ |
| 56.768.25-56.ko.1.8 | $56$ | $2$ | $2$ | $25$ | $4$ | $2^{3}\cdot8$ |
| 56.768.25-56.ko.1.13 | $56$ | $2$ | $2$ | $25$ | $4$ | $2^{3}\cdot8$ |
| 56.768.25-56.ko.2.12 | $56$ | $2$ | $2$ | $25$ | $4$ | $2^{3}\cdot8$ |
| 56.768.25-56.ko.2.13 | $56$ | $2$ | $2$ | $25$ | $4$ | $2^{3}\cdot8$ |
| 56.768.25-56.kr.1.7 | $56$ | $2$ | $2$ | $25$ | $6$ | $2^{3}\cdot8$ |
| 56.768.25-56.kr.1.14 | $56$ | $2$ | $2$ | $25$ | $6$ | $2^{3}\cdot8$ |
| 56.768.25-56.kr.2.11 | $56$ | $2$ | $2$ | $25$ | $6$ | $2^{3}\cdot8$ |
| 56.768.25-56.kr.2.14 | $56$ | $2$ | $2$ | $25$ | $6$ | $2^{3}\cdot8$ |
| 56.768.25-56.ku.1.1 | $56$ | $2$ | $2$ | $25$ | $9$ | $1^{10}\cdot2^{2}$ |
| 56.768.25-56.kz.1.1 | $56$ | $2$ | $2$ | $25$ | $5$ | $1^{10}\cdot2^{2}$ |
| 56.768.25-56.lp.1.5 | $56$ | $2$ | $2$ | $25$ | $10$ | $1^{10}\cdot2^{2}$ |
| 56.768.25-56.lv.1.5 | $56$ | $2$ | $2$ | $25$ | $10$ | $1^{10}\cdot2^{2}$ |
| 56.1152.31-56.hj.1.15 | $56$ | $3$ | $3$ | $31$ | $6$ | $2^{10}$ |
| 56.1152.31-56.hj.2.8 | $56$ | $3$ | $3$ | $31$ | $6$ | $2^{10}$ |
| 56.1152.31-56.ie.1.7 | $56$ | $3$ | $3$ | $31$ | $14$ | $1^{20}$ |
| 56.2688.89-56.qn.1.6 | $56$ | $7$ | $7$ | $89$ | $33$ | $1^{48}\cdot2^{15}$ |