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: | $\begin{bmatrix}103&216\\216&159\end{bmatrix}$, $\begin{bmatrix}142&199\\215&222\end{bmatrix}$, $\begin{bmatrix}152&39\\73&254\end{bmatrix}$, $\begin{bmatrix}240&11\\197&98\end{bmatrix}$, $\begin{bmatrix}251&184\\58&229\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.12.0.z.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 1270 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{16}\cdot5}\cdot\frac{x^{12}(25x^{4}-1280x^{2}y^{2}+4096y^{4})^{3}}{y^{8}x^{14}(5x^{2}-256y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 28.12.0-4.c.1.2 | $28$ | $2$ | $2$ | $0$ | $0$ |
| 280.12.0-4.c.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 |
|---|---|---|---|---|
| 280.48.0-40.m.1.12 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.o.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.w.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.x.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.bn.1.11 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.bo.1.4 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.bs.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.bv.1.4 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.cn.1.16 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.cp.1.11 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.cr.1.16 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.ct.1.11 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.dl.1.16 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.dn.1.12 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.dp.1.16 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.dr.1.14 | $280$ | $2$ | $2$ | $0$ |
| 280.120.4-40.bn.1.12 | $280$ | $5$ | $5$ | $4$ |
| 280.144.3-40.bz.1.10 | $280$ | $6$ | $6$ | $3$ |
| 280.192.5-280.bx.1.61 | $280$ | $8$ | $8$ | $5$ |
| 280.240.7-40.cl.1.11 | $280$ | $10$ | $10$ | $7$ |
| 280.504.16-280.cv.1.52 | $280$ | $21$ | $21$ | $16$ |