Invariants
| Level: | $280$ | $\SL_2$-level: | $56$ | Newform level: | $112$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $4 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $2$ are rational) | Cusp widths | $1^{4}\cdot7^{4}\cdot8\cdot56$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 56C4 |
Level structure
| $\GL_2(\Z/280\Z)$-generators: | $\begin{bmatrix}102&57\\31&268\end{bmatrix}$, $\begin{bmatrix}109&196\\136&71\end{bmatrix}$, $\begin{bmatrix}129&170\\134&249\end{bmatrix}$, $\begin{bmatrix}151&174\\246&107\end{bmatrix}$, $\begin{bmatrix}266&227\\53&244\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.96.4.d.1 for the level structure with $-I$) |
| Cyclic 280-isogeny field degree: | $24$ |
| Cyclic 280-torsion field degree: | $2304$ |
| Full 280-torsion field degree: | $7741440$ |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 14 x^{2} - z w $ |
| $=$ | $7 x z^{2} + x w^{2} + 14 y^{3} - 5 y z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{5} - 10 x^{2} y z^{2} + 28 x z^{4} + 2 y^{3} z^{2} $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:1)$, $(0:0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2744xyz^{14}+820694xyz^{12}w^{2}+26080866xyz^{10}w^{4}+109296950xyz^{8}w^{6}+74037530xyz^{6}w^{8}+10159758xyz^{4}w^{10}+241178xyz^{2}w^{12}+392xyw^{14}-14588y^{2}z^{13}w-1446522y^{2}z^{11}w^{3}-22384362y^{2}z^{9}w^{5}-41156570y^{2}z^{7}w^{7}-12989802y^{2}z^{5}w^{9}-742938y^{2}z^{3}w^{11}-4508y^{2}zw^{13}-49z^{16}-17828z^{14}w^{2}-1051096z^{12}w^{4}-6899866z^{10}w^{6}-6426875z^{8}w^{8}-1529626z^{6}w^{10}-120904z^{4}w^{12}-1988z^{2}w^{14}-w^{16}}{wz^{9}(546xyz^{3}w+462xyzw^{3}-14y^{2}z^{4}-350y^{2}z^{2}w^{2}-14y^{2}w^{4}-22z^{5}w-45z^{3}w^{3}-6zw^{5})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 56.96.4.d.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{14}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{5}-10X^{2}YZ^{2}+2Y^{3}Z^{2}+28XZ^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 140.96.2-28.b.1.2 | $140$ | $2$ | $2$ | $2$ | $?$ |
| 280.96.2-28.b.1.15 | $280$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 280.384.11-56.k.2.9 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-56.bu.1.6 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-56.ci.2.7 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-56.co.2.5 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-56.ec.2.5 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-56.ef.2.5 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-56.eg.2.7 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-56.ej.2.5 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-280.gp.2.4 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-280.gr.1.16 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-280.gt.2.4 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-280.gv.2.6 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-280.kl.2.4 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-280.kn.2.12 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-280.kp.2.4 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-280.kr.2.6 | $280$ | $2$ | $2$ | $11$ |