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: | $4$ (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: | $\begin{bmatrix}187&216\\245&249\end{bmatrix}$, $\begin{bmatrix}207&212\\62&163\end{bmatrix}$, $\begin{bmatrix}233&186\\149&101\end{bmatrix}$, $\begin{bmatrix}243&224\\256&279\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.12.0.g.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 28 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^7}{5^2}\cdot\frac{(5x+8y)^{12}(75x^{4}+2400x^{3}y-4480x^{2}y^{2}-51200xy^{3}-53248y^{4})^{3}}{(5x+8y)^{12}(5x^{2}-80xy-192y^{2})^{2}(5x^{2}+16xy+64y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 56.12.0-4.a.1.1 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 140.12.0-4.a.1.2 | $140$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 280.120.4-40.p.1.4 | $280$ | $5$ | $5$ | $4$ |
| 280.144.3-40.v.1.4 | $280$ | $6$ | $6$ | $3$ |
| 280.240.7-40.bb.1.7 | $280$ | $10$ | $10$ | $7$ |
| 280.192.5-280.m.1.17 | $280$ | $8$ | $8$ | $5$ |
| 280.504.16-280.y.1.13 | $280$ | $21$ | $21$ | $16$ |