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 | $2^{2}\cdot4^{2}$ | 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: | 4E0 |
Level structure
| $\GL_2(\Z/280\Z)$-generators: | $\begin{bmatrix}67&192\\257&143\end{bmatrix}$, $\begin{bmatrix}117&80\\275&27\end{bmatrix}$, $\begin{bmatrix}121&124\\221&237\end{bmatrix}$, $\begin{bmatrix}247&188\\17&227\end{bmatrix}$, $\begin{bmatrix}261&44\\6&47\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.12.0.h.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 480 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^6}{5}\cdot\frac{x^{12}(11x^{4}+192x^{3}y-4736x^{2}y^{2}+12288xy^{3}+45056y^{4})^{3}}{x^{12}(x-8y)^{2}(x+8y)^{2}(3x^{2}-32xy+192y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 56.12.0-4.c.1.4 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 280.12.0-4.c.1.4 | $280$ | $2$ | $2$ | $0$ | $?$ |
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.bm.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.bm.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.bn.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.bn.1.9 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.bw.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.bw.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.bx.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.bx.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.ce.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.ce.1.15 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.cf.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.cf.1.13 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.ci.1.7 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.ci.1.9 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.cj.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.cj.1.14 | $280$ | $2$ | $2$ | $0$ |
| 280.120.4-20.l.1.3 | $280$ | $5$ | $5$ | $4$ |
| 280.144.3-20.p.1.10 | $280$ | $6$ | $6$ | $3$ |
| 280.192.5-140.l.1.19 | $280$ | $8$ | $8$ | $5$ |
| 280.240.7-20.t.1.7 | $280$ | $10$ | $10$ | $7$ |
| 280.504.16-140.t.1.9 | $280$ | $21$ | $21$ | $16$ |