Invariants
| Level: | $280$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot4$ | ||
| 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: | 8G0 |
Level structure
| $\GL_2(\Z/280\Z)$-generators: | $\begin{bmatrix}5&38\\26&271\end{bmatrix}$, $\begin{bmatrix}67&226\\176&101\end{bmatrix}$, $\begin{bmatrix}101&170\\186&59\end{bmatrix}$, $\begin{bmatrix}127&216\\128&23\end{bmatrix}$, $\begin{bmatrix}179&180\\186&11\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.24.0.k.1 for the level structure with $-I$) |
| Cyclic 280-isogeny field degree: | $192$ |
| Cyclic 280-torsion field degree: | $18432$ |
| Full 280-torsion field degree: | $30965760$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 16 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{7^4}\cdot\frac{(3x-2y)^{24}(39x^{4}-168x^{3}y+280x^{2}y^{2}-196xy^{3}+49y^{4})^{3}(67x^{4}-224x^{3}y+308x^{2}y^{2}-196xy^{3}+49y^{4})^{3}}{x^{8}(x-y)^{8}(3x-2y)^{24}(53x^{4}-196x^{3}y+294x^{2}y^{2}-196xy^{3}+49y^{4})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.24.0-4.a.1.3 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 140.24.0-4.a.1.1 | $140$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 280.96.1-56.b.1.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-56.d.1.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-56.e.1.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-56.h.1.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-56.j.1.4 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-56.l.1.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-56.o.1.5 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-56.z.1.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.cj.1.6 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.cl.1.7 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.cn.1.6 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.cp.1.7 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.cr.1.3 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.ct.1.7 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.cv.1.4 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.cx.1.7 | $280$ | $2$ | $2$ | $1$ |
| 280.240.8-280.bj.1.9 | $280$ | $5$ | $5$ | $8$ |
| 280.288.7-280.cf.1.5 | $280$ | $6$ | $6$ | $7$ |
| 280.384.11-56.bh.1.5 | $280$ | $8$ | $8$ | $11$ |
| 280.480.15-280.cv.1.11 | $280$ | $10$ | $10$ | $15$ |