Invariants
| Level: | $280$ | $\SL_2$-level: | $56$ | Newform level: | $448$ | ||
| 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 | $1^{2}\cdot4^{2}$ | ||
| 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: | $\begin{bmatrix}16&41\\53&214\end{bmatrix}$, $\begin{bmatrix}20&159\\67&238\end{bmatrix}$, $\begin{bmatrix}97&150\\210&23\end{bmatrix}$, $\begin{bmatrix}154&3\\127&128\end{bmatrix}$, $\begin{bmatrix}256&103\\103&102\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.96.4.c.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 $ | $=$ | $ 7 x^{2} - z w $ |
| $=$ | $7 x z^{2} + x w^{2} + 7 y^{3} - 5 y z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{5} - 5 x^{2} y z^{2} + 7 x z^{4} + 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1372xyz^{14}+410347xyz^{12}w^{2}+13040433xyz^{10}w^{4}+54648475xyz^{8}w^{6}+37018765xyz^{6}w^{8}+5079879xyz^{4}w^{10}+120589xyz^{2}w^{12}+196xyw^{14}-7294y^{2}z^{13}w-723261y^{2}z^{11}w^{3}-11192181y^{2}z^{9}w^{5}-20578285y^{2}z^{7}w^{7}-6494901y^{2}z^{5}w^{9}-371469y^{2}z^{3}w^{11}-2254y^{2}zw^{13}-49z^{16}-17828z^{14}w^{2}-1051096z^{12}w^{4}-6899866z^{10}w^{6}-6426875z^{8}w^{8}-1529626z^{6}w^{10}-120904z^{4}w^{12}-1988z^{2}w^{14}-w^{16}}{wz^{9}(273xyz^{3}w+231xyzw^{3}-7y^{2}z^{4}-175y^{2}z^{2}w^{2}-7y^{2}w^{4}-22z^{5}w-45z^{3}w^{3}-6zw^{5})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 56.96.4.c.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{7}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{5}-5X^{2}YZ^{2}+Y^{3}Z^{2}+7XZ^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 280.96.2-28.b.1.3 | $280$ | $2$ | $2$ | $2$ | $?$ |
| 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.l.2.5 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-56.bo.2.6 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-56.cj.2.5 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-56.cn.2.7 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-56.ed.2.3 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-56.ee.2.5 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-56.eh.2.5 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-56.ei.2.7 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-280.go.2.3 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-280.gq.2.14 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-280.gs.2.3 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-280.gu.2.4 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-280.kk.2.3 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-280.km.2.6 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-280.ko.2.3 | $280$ | $2$ | $2$ | $11$ |
| 280.384.11-280.kq.2.4 | $280$ | $2$ | $2$ | $11$ |