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: | $4$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot2\cdot8$ | 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: | 8C0 |
Level structure
| $\GL_2(\Z/280\Z)$-generators: | $\begin{bmatrix}0&201\\13&248\end{bmatrix}$, $\begin{bmatrix}26&45\\209&242\end{bmatrix}$, $\begin{bmatrix}53&170\\86&89\end{bmatrix}$, $\begin{bmatrix}124&5\\219&278\end{bmatrix}$, $\begin{bmatrix}279&30\\220&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.12.0.y.1 for the level structure with $-I$) |
| Cyclic 280-isogeny field degree: | $96$ |
| Cyclic 280-torsion field degree: | $9216$ |
| 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 774 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^6}{3^2\cdot7^4}\cdot\frac{x^{12}(49x^{4}-504x^{2}y^{2}+324y^{4})^{3}}{y^{2}x^{20}(14x^{2}-9y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.12.0-4.c.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 280.12.0-4.c.1.5 | $280$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 280.48.0-56.l.1.12 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-56.n.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-56.t.1.4 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-56.v.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-56.bi.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-56.bl.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-56.bn.1.6 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-56.bo.1.6 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.bw.1.11 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.by.1.10 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.ce.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.cg.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.dk.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.dn.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.dt.1.9 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.du.1.9 | $280$ | $2$ | $2$ | $0$ |
| 280.120.4-280.bw.1.19 | $280$ | $5$ | $5$ | $4$ |
| 280.144.3-280.ci.1.26 | $280$ | $6$ | $6$ | $3$ |
| 280.192.5-56.bm.1.21 | $280$ | $8$ | $8$ | $5$ |
| 280.240.7-280.cu.1.38 | $280$ | $10$ | $10$ | $7$ |
| 280.504.16-56.ck.1.6 | $280$ | $21$ | $21$ | $16$ |