Invariants
| Level: | $280$ | $\SL_2$-level: | $4$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4G0 |
Level structure
| $\GL_2(\Z/280\Z)$-generators: | $\begin{bmatrix}3&16\\60&57\end{bmatrix}$, $\begin{bmatrix}37&180\\104&37\end{bmatrix}$, $\begin{bmatrix}79&134\\208&247\end{bmatrix}$, $\begin{bmatrix}157&158\\0&247\end{bmatrix}$, $\begin{bmatrix}205&66\\48&133\end{bmatrix}$, $\begin{bmatrix}271&170\\40&67\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 28.24.0.c.1 for the level structure with $-I$) |
| Cyclic 280-isogeny field degree: | $96$ |
| Cyclic 280-torsion field degree: | $9216$ |
| Full 280-torsion field degree: | $30965760$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 55 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{7^2}\cdot\frac{(x-3y)^{24}(4033x^{8}-49952x^{7}y+285376x^{6}y^{2}-1492736x^{5}y^{3}+13006560x^{4}y^{4}-57777664x^{3}y^{5}+120472576x^{2}y^{6}-127848448xy^{7}+118628608y^{8})^{3}}{(x-3y)^{24}(x+2y)^{4}(3x-14y)^{4}(x^{2}-56xy+84y^{2})^{4}(2x^{2}-7xy+28y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.24.0-4.b.1.6 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 280.24.0-28.a.1.2 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.24.0-28.a.1.4 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.24.0-4.b.1.1 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.24.0-28.b.1.1 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.24.0-28.b.1.3 | $280$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.