Invariants
| Level: | $280$ | $\SL_2$-level: | $4$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| 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: | 4E0 |
Level structure
| $\GL_2(\Z/280\Z)$-generators: | $\begin{bmatrix}71&264\\140&153\end{bmatrix}$, $\begin{bmatrix}85&138\\218&119\end{bmatrix}$, $\begin{bmatrix}225&242\\242&173\end{bmatrix}$, $\begin{bmatrix}271&260\\6&87\end{bmatrix}$, $\begin{bmatrix}277&184\\146&81\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 28.12.0.a.1 for the level structure with $-I$) |
| Cyclic 280-isogeny field degree: | $192$ |
| Cyclic 280-torsion field degree: | $18432$ |
| Full 280-torsion field degree: | $61931520$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 556 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3^4\cdot7^2}\cdot\frac{x^{12}(784x^{4}-252x^{2}y^{2}+81y^{4})^{3}}{y^{4}x^{16}(28x^{2}-9y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.12.0-2.a.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 280.12.0-2.a.1.1 | $280$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 280.48.0-28.a.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-28.a.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-28.c.1.4 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-28.c.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.192.5-28.c.1.4 | $280$ | $8$ | $8$ | $5$ |
| 280.504.16-28.c.1.5 | $280$ | $21$ | $21$ | $16$ |
| 280.48.0-56.b.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-56.b.1.7 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-56.f.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-56.f.1.7 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-140.d.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-140.d.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-140.f.1.4 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-140.f.1.8 | $280$ | $2$ | $2$ | $0$ |
| 280.120.4-140.a.1.8 | $280$ | $5$ | $5$ | $4$ |
| 280.144.3-140.a.1.7 | $280$ | $6$ | $6$ | $3$ |
| 280.240.7-140.a.1.13 | $280$ | $10$ | $10$ | $7$ |
| 280.48.0-280.i.1.4 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.i.1.16 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.o.1.4 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.o.1.16 | $280$ | $2$ | $2$ | $0$ |