Invariants
| Level: | $56$ | $\SL_2$-level: | $28$ | 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 | $14^{12}$ | 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: | 14B9 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.336.9.80 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}7&48\\22&35\end{bmatrix}$, $\begin{bmatrix}31&3\\12&25\end{bmatrix}$, $\begin{bmatrix}45&17\\10&25\end{bmatrix}$, $\begin{bmatrix}53&14\\28&25\end{bmatrix}$, $\begin{bmatrix}53&20\\34&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 28.168.9.a.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 $ | $=$ | $ x v + x r - z v $ |
| $=$ | $x r + x s - y v$ | |
| $=$ | $y v + y r - z r - z s$ | |
| $=$ | $x r - y r + z s + w v + w r - w s - t v - 2 t r - t s - u v - u r + u s$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 32 x^{16} - 208 x^{15} y - 512 x^{14} y^{2} - 512 x^{14} z^{2} - 560 x^{13} y^{3} - 3412 x^{13} y z^{2} + \cdots + 4 y^{8} z^{8} $ |
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 |
|---|
| $(1:-2:-1:1:2:1:0:0:0)$ |
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 :
| $\displaystyle X$ | $=$ | $\displaystyle -2v-r+4s$ |
| $\displaystyle Y$ | $=$ | $\displaystyle v-3r-2s$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -2v-r-3s$ |
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.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle v$ |
Equation of the image curve:
| $0$ | $=$ | $ -32X^{16}-208X^{15}Y-512X^{14}Y^{2}-512X^{14}Z^{2}-560X^{13}Y^{3}-3412X^{13}YZ^{2}-144X^{12}Y^{4}-8408X^{12}Y^{2}Z^{2}-255X^{12}Z^{4}+272X^{11}Y^{5}-9064X^{11}Y^{3}Z^{2}-3684X^{11}YZ^{4}+288X^{10}Y^{6}-2192X^{10}Y^{4}Z^{2}-11126X^{10}Y^{2}Z^{4}+250X^{10}Z^{6}+112X^{9}Y^{7}+4396X^{9}Y^{5}Z^{2}-13028X^{9}Y^{3}Z^{4}-160X^{9}YZ^{6}+16X^{8}Y^{8}+4504X^{8}Y^{6}Z^{2}-4163X^{8}Y^{4}Z^{4}-1512X^{8}Y^{2}Z^{6}+25X^{8}Z^{8}+1712X^{7}Y^{7}Z^{2}+4444X^{7}Y^{5}Z^{4}-1394X^{7}Y^{3}Z^{6}+320X^{7}YZ^{8}+240X^{6}Y^{8}Z^{2}+4820X^{6}Y^{6}Z^{4}+219X^{6}Y^{4}Z^{6}+1394X^{6}Y^{2}Z^{8}+1788X^{5}Y^{7}Z^{4}+896X^{5}Y^{5}Z^{6}+2518X^{5}Y^{3}Z^{8}+244X^{4}Y^{8}Z^{4}+481X^{4}Y^{6}Z^{6}+2349X^{4}Y^{4}Z^{8}+104X^{3}Y^{7}Z^{6}+1238X^{3}Y^{5}Z^{8}+8X^{2}Y^{8}Z^{6}+373X^{2}Y^{6}Z^{8}+60XY^{7}Z^{8}+4Y^{8}Z^{8} $ |
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.a.1.1 | $56$ | $28$ | $28$ | $0$ | $0$ | 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.c.1.1 | $56$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 56.672.17-28.d.1.5 | $56$ | $2$ | $2$ | $17$ | $2$ | $1^{8}$ |
| 56.672.17-28.e.1.2 | $56$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 56.672.17-28.f.1.2 | $56$ | $2$ | $2$ | $17$ | $4$ | $1^{8}$ |
| 56.672.21-28.i.1.1 | $56$ | $2$ | $2$ | $21$ | $1$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-28.j.1.3 | $56$ | $2$ | $2$ | $21$ | $4$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-28.k.1.4 | $56$ | $2$ | $2$ | $21$ | $2$ | $1^{10}\cdot2$ |
| 56.672.21-28.l.1.3 | $56$ | $2$ | $2$ | $21$ | $3$ | $1^{10}\cdot2$ |
| 56.672.21-28.m.1.3 | $56$ | $2$ | $2$ | $21$ | $4$ | $1^{10}\cdot2$ |
| 56.672.21-28.n.1.3 | $56$ | $2$ | $2$ | $21$ | $5$ | $1^{10}\cdot2$ |
| 56.672.21-28.o.1.2 | $56$ | $2$ | $2$ | $21$ | $3$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-28.p.1.1 | $56$ | $2$ | $2$ | $21$ | $5$ | $1^{6}\cdot2^{3}$ |
| 56.1008.25-28.c.1.2 | $56$ | $3$ | $3$ | $25$ | $3$ | $1^{10}\cdot2^{3}$ |
| 56.672.17-56.m.1.5 | $56$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 56.672.17-56.p.1.2 | $56$ | $2$ | $2$ | $17$ | $4$ | $1^{8}$ |
| 56.672.17-56.s.1.1 | $56$ | $2$ | $2$ | $17$ | $2$ | $1^{8}$ |
| 56.672.17-56.v.1.6 | $56$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 56.672.21-56.y.1.5 | $56$ | $2$ | $2$ | $21$ | $6$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.bb.1.5 | $56$ | $2$ | $2$ | $21$ | $8$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.be.1.2 | $56$ | $2$ | $2$ | $21$ | $4$ | $1^{10}\cdot2$ |
| 56.672.21-56.bh.1.2 | $56$ | $2$ | $2$ | $21$ | $5$ | $1^{10}\cdot2$ |
| 56.672.21-56.bk.1.6 | $56$ | $2$ | $2$ | $21$ | $6$ | $1^{10}\cdot2$ |
| 56.672.21-56.bn.1.4 | $56$ | $2$ | $2$ | $21$ | $7$ | $1^{10}\cdot2$ |
| 56.672.21-56.bq.1.3 | $56$ | $2$ | $2$ | $21$ | $4$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.bt.1.5 | $56$ | $2$ | $2$ | $21$ | $5$ | $1^{6}\cdot2^{3}$ |