Invariants
| Level: | $280$ | $\SL_2$-level: | $8$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $2^{2}$ | ||
| 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: | 4E0 |
Level structure
| $\GL_2(\Z/280\Z)$-generators: | $\begin{bmatrix}35&248\\278&39\end{bmatrix}$, $\begin{bmatrix}85&256\\206&217\end{bmatrix}$, $\begin{bmatrix}147&44\\146&15\end{bmatrix}$, $\begin{bmatrix}171&94\\224&57\end{bmatrix}$, $\begin{bmatrix}227&160\\257&91\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.12.0.e.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 150 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^{14}}{5}\cdot\frac{(2x-y)^{12}(16x^{4}-4x^{3}y+11x^{2}y^{2}+xy^{3}+y^{4})^{3}}{(2x-y)^{12}(4x^{2}+y^{2})^{2}(4x^{2}+2xy-y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 56.12.0-4.b.1.3 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 280.12.0-4.b.1.3 | $280$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 280.48.0-20.e.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-20.e.1.6 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-20.f.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-20.f.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-140.i.1.8 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-140.i.1.12 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-140.j.1.14 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-140.j.1.15 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.bq.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.bq.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.br.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.br.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.bu.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.bu.1.9 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.bv.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.bv.1.9 | $280$ | $2$ | $2$ | $0$ |
| 280.120.4-20.i.1.7 | $280$ | $5$ | $5$ | $4$ |
| 280.144.3-20.m.1.8 | $280$ | $6$ | $6$ | $3$ |
| 280.192.5-140.i.1.14 | $280$ | $8$ | $8$ | $5$ |
| 280.240.7-20.q.1.15 | $280$ | $10$ | $10$ | $7$ |
| 280.504.16-140.q.1.19 | $280$ | $21$ | $21$ | $16$ |