Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $784$ | ||
| Index: | $336$ | $\PSL_2$-index: | $168$ | ||||
| Genus: | $9 = 1 + \frac{ 168 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $1$ is rational) | Cusp widths | $7^{8}\cdot28^{4}$ | Cusp orbits | $1\cdot2\cdot3\cdot6$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 6$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 28A9 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.336.9.75 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}9&0\\52&33\end{bmatrix}$, $\begin{bmatrix}17&21\\36&53\end{bmatrix}$, $\begin{bmatrix}31&36\\22&1\end{bmatrix}$, $\begin{bmatrix}45&45\\52&53\end{bmatrix}$, $\begin{bmatrix}51&37\\16&19\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 28.168.9.b.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $8$ |
| Cyclic 56-torsion field degree: | $192$ |
| Full 56-torsion field degree: | $9216$ |
Jacobian
| Conductor: | $2^{27}\cdot7^{16}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2^{3}$ |
| Newforms: | 14.2.a.a, 98.2.a.b, 112.2.a.c, 784.2.a.d, 784.2.a.l, 784.2.a.m |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ y v + t r $ |
| $=$ | $x v + t v + t r + t s$ | |
| $=$ | $x r - y v - y r - y s$ | |
| $=$ | $x v + x s + z v + 2 z r + z s + w v + w s + t r + u v + u s$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 14700 x^{16} - 544880 x^{15} y + 4708508 x^{14} y^{2} + 3724 x^{14} z^{2} - 9545004 x^{13} y^{3} + \cdots + 400 y^{10} z^{6} $ |
Rational points
This modular curve has 1 rational cusp but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(-2:1:1:1:1:1:0:0:0)$ |
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 14.84.3.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -v+4r-s$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 4v-2r-3s$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -v-3r-s$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{2}Y^{2}+X^{3}Z+4XY^{2}Z+Y^{3}Z+2X^{2}Z^{2}-XYZ^{2}-2XZ^{3} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 28.168.9.b.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 7v$ |
Equation of the image curve:
| $0$ | $=$ | $ 14700X^{16}-544880X^{15}Y+4708508X^{14}Y^{2}+3724X^{14}Z^{2}-9545004X^{13}Y^{3}-60760X^{13}YZ^{2}-23422441X^{12}Y^{4}+408268X^{12}Y^{2}Z^{2}+196X^{12}Z^{4}+94646048X^{11}Y^{5}-1658944X^{11}Y^{3}Z^{2}-3136X^{11}YZ^{4}+2339946X^{10}Y^{6}+2830828X^{10}Y^{4}Z^{2}+24892X^{10}Y^{2}Z^{4}+4X^{10}Z^{6}-261537353X^{9}Y^{7}+8157128X^{9}Y^{5}Z^{2}-128772X^{9}Y^{3}Z^{4}-88X^{9}YZ^{6}+76905647X^{8}Y^{8}-28865312X^{8}Y^{6}Z^{2}+419881X^{8}Y^{4}Z^{4}+636X^{8}Y^{2}Z^{6}+335938561X^{7}Y^{9}-1545460X^{7}Y^{7}Z^{2}-813890X^{7}Y^{5}Z^{4}-1144X^{7}Y^{3}Z^{6}-24537877X^{6}Y^{10}+41913816X^{6}Y^{8}Z^{2}+567665X^{6}Y^{6}Z^{4}-4788X^{6}Y^{4}Z^{6}-140003682X^{5}Y^{11}-17966144X^{5}Y^{9}Z^{2}+1611806X^{5}Y^{7}Z^{4}+14616X^{5}Y^{5}Z^{6}-18288760X^{4}Y^{12}-19717404X^{4}Y^{10}Z^{2}-1010576X^{4}Y^{8}Z^{4}+10248X^{4}Y^{6}Z^{6}-1425851X^{3}Y^{13}+14508312X^{3}Y^{11}Z^{2}-169050X^{3}Y^{9}Z^{4}-29504X^{3}Y^{7}Z^{6}-212513X^{2}Y^{14}+2731456X^{2}Y^{12}Z^{2}+456925X^{2}Y^{10}Z^{4}+5956X^{2}Y^{8}Z^{6}+9849XY^{15}-2204412XY^{13}Z^{2}-183750XY^{11}Z^{4}+4240XY^{9}Z^{6}-98Y^{16}-350840Y^{14}Z^{2}+30625Y^{12}Z^{4}+400Y^{10}Z^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.12.0-4.b.1.3 | $56$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.672.17-28.g.1.4 | $56$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 56.672.17-28.h.1.3 | $56$ | $2$ | $2$ | $17$ | $4$ | $1^{8}$ |
| 56.672.17-28.k.1.2 | $56$ | $2$ | $2$ | $17$ | $2$ | $1^{8}$ |
| 56.672.17-28.l.1.1 | $56$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 56.672.17-56.y.1.2 | $56$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 56.672.17-56.bb.1.2 | $56$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 56.672.17-56.bk.1.3 | $56$ | $2$ | $2$ | $17$ | $4$ | $1^{8}$ |
| 56.672.17-56.bn.1.3 | $56$ | $2$ | $2$ | $17$ | $2$ | $1^{8}$ |
| 56.672.21-28.a.1.15 | $56$ | $2$ | $2$ | $21$ | $5$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-28.i.1.6 | $56$ | $2$ | $2$ | $21$ | $1$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.i.1.5 | $56$ | $2$ | $2$ | $21$ | $5$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-28.q.1.2 | $56$ | $2$ | $2$ | $21$ | $3$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-28.r.1.2 | $56$ | $2$ | $2$ | $21$ | $4$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-28.u.1.5 | $56$ | $2$ | $2$ | $21$ | $4$ | $1^{10}\cdot2$ |
| 56.672.21-28.v.1.5 | $56$ | $2$ | $2$ | $21$ | $3$ | $1^{10}\cdot2$ |
| 56.672.21-28.y.1.5 | $56$ | $2$ | $2$ | $21$ | $5$ | $1^{10}\cdot2$ |
| 56.672.21-28.z.1.6 | $56$ | $2$ | $2$ | $21$ | $2$ | $1^{10}\cdot2$ |
| 56.672.21-56.z.1.5 | $56$ | $2$ | $2$ | $21$ | $6$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.bw.1.7 | $56$ | $2$ | $2$ | $21$ | $4$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.bz.1.7 | $56$ | $2$ | $2$ | $21$ | $8$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.cy.1.8 | $56$ | $2$ | $2$ | $21$ | $6$ | $1^{10}\cdot2$ |
| 56.672.21-56.db.1.8 | $56$ | $2$ | $2$ | $21$ | $5$ | $1^{10}\cdot2$ |
| 56.672.21-56.dk.1.8 | $56$ | $2$ | $2$ | $21$ | $7$ | $1^{10}\cdot2$ |
| 56.672.21-56.dn.1.8 | $56$ | $2$ | $2$ | $21$ | $4$ | $1^{10}\cdot2$ |
| 56.1008.25-28.e.1.5 | $56$ | $3$ | $3$ | $25$ | $3$ | $1^{10}\cdot2^{3}$ |