Invariants
| Level: | $280$ | $\SL_2$-level: | $8$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| 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: | 8O0 |
Level structure
| $\GL_2(\Z/280\Z)$-generators: | $\begin{bmatrix}57&120\\80&189\end{bmatrix}$, $\begin{bmatrix}141&32\\27&183\end{bmatrix}$, $\begin{bmatrix}141&192\\12&125\end{bmatrix}$, $\begin{bmatrix}197&104\\178&189\end{bmatrix}$, $\begin{bmatrix}279&152\\123&221\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.0.bd.1 for the level structure with $-I$) |
| Cyclic 280-isogeny field degree: | $48$ |
| Cyclic 280-torsion field degree: | $4608$ |
| Full 280-torsion field degree: | $15482880$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 5 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^6\cdot3^4\cdot7^2}\cdot\frac{x^{48}(2401x^{8}-395136x^{6}y^{2}+2032128x^{4}y^{4}+167215104x^{2}y^{6}+429981696y^{8})^{3}(2401x^{8}+395136x^{6}y^{2}+2032128x^{4}y^{4}-167215104x^{2}y^{6}+429981696y^{8})^{3}}{y^{4}x^{52}(7x^{2}-144y^{2})^{2}(7x^{2}+144y^{2})^{2}(49x^{4}+20736y^{4})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.48.0-8.q.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 280.48.0-8.q.1.5 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.bu.1.3 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.bu.1.16 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.bv.2.1 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.bv.2.14 | $280$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 280.192.1-56.cj.2.2 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.ck.1.3 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.cl.1.7 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.cm.2.3 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.pp.1.8 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.pq.1.8 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.pt.1.14 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.pv.1.8 | $280$ | $2$ | $2$ | $1$ |
| 280.480.16-280.ef.2.13 | $280$ | $5$ | $5$ | $16$ |