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: | $10$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $2^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8O0 |
Level structure
| $\GL_2(\Z/280\Z)$-generators: | $\begin{bmatrix}5&244\\28&47\end{bmatrix}$, $\begin{bmatrix}17&192\\278&87\end{bmatrix}$, $\begin{bmatrix}147&228\\246&203\end{bmatrix}$, $\begin{bmatrix}215&136\\12&279\end{bmatrix}$, $\begin{bmatrix}237&196\\154&275\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.0.r.1 for the level structure with $-I$) |
| Cyclic 280-isogeny field degree: | $96$ |
| Cyclic 280-torsion field degree: | $9216$ |
| 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 4 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4\cdot7^4}\cdot\frac{(x+2y)^{48}(3361x^{16}-26656x^{15}y+1154272x^{14}y^{2}+8661632x^{13}y^{3}-2888256x^{12}y^{4}-248496640x^{11}y^{5}+10335879680x^{10}y^{6}+772007936x^{9}y^{7}+5511005696x^{8}y^{8}-21616222208x^{7}y^{9}+8103329669120x^{6}y^{10}+5454998241280x^{5}y^{11}-1775283879936x^{4}y^{12}-149069874200576x^{3}y^{13}+556232484978688x^{2}y^{14}+359667502415872xy^{15}+1269792516407296y^{16})^{3}}{(x+2y)^{48}(x^{2}+28y^{2})^{8}(x^{2}-4xy-28y^{2})^{8}(x^{2}+28xy-28y^{2})^{4}(3x^{2}-28xy+140y^{2})^{2}(5x^{2}+28xy+84y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.48.0-8.e.1.15 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 280.48.0-8.e.1.6 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.i.1.21 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.i.1.28 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.l.1.4 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.l.1.17 | $280$ | $2$ | $2$ | $0$ | $?$ |
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.r.1.2 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.u.1.5 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.w.1.5 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.z.2.1 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.be.1.4 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.bf.1.7 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.bg.1.6 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.bh.1.3 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.im.2.2 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.in.2.14 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.iq.1.8 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.ir.1.2 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.js.1.6 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.jt.1.6 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.jw.2.6 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.jx.2.6 | $280$ | $2$ | $2$ | $1$ |
| 280.480.16-280.cz.2.15 | $280$ | $5$ | $5$ | $16$ |