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: | $6$ (none of which are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4G0 |
Level structure
| $\GL_2(\Z/280\Z)$-generators: | $\begin{bmatrix}131&122\\10&91\end{bmatrix}$, $\begin{bmatrix}183&214\\270&241\end{bmatrix}$, $\begin{bmatrix}207&224\\8&129\end{bmatrix}$, $\begin{bmatrix}269&134\\212&231\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.24.0.b.1 for the level structure with $-I$) |
| Cyclic 280-isogeny field degree: | $192$ |
| Cyclic 280-torsion field degree: | $18432$ |
| 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 10 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^6}{3^4\cdot7^2}\cdot\frac{(3x-4y)^{24}(179948547x^{8}-9827643168x^{7}y+237212587584x^{6}y^{2}-3244257805824x^{5}y^{3}+27548146967040x^{4}y^{4}-152090539646976x^{3}y^{5}+555452373417984x^{2}y^{6}-1278824674885632xy^{7}+1442575327363072y^{8})^{3}}{(3x-4y)^{24}(9x^{2}+216xy-1904y^{2})^{4}(27x^{2}-392xy+1008y^{2})^{4}(117x^{2}-1512xy+6608y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.24.0-8.a.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 140.24.0-28.a.1.3 | $140$ | $2$ | $2$ | $0$ | $?$ |
| 280.24.0-8.a.1.4 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.24.0-28.a.1.1 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.24.0-56.a.1.1 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.24.0-56.a.1.8 | $280$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 280.240.8-280.b.1.10 | $280$ | $5$ | $5$ | $8$ |
| 280.288.7-280.d.1.21 | $280$ | $6$ | $6$ | $7$ |
| 280.384.11-56.e.1.7 | $280$ | $8$ | $8$ | $11$ |
| 280.480.15-280.b.1.17 | $280$ | $10$ | $10$ | $15$ |