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: | $\begin{bmatrix}67&76\\156&239\end{bmatrix}$, $\begin{bmatrix}77&124\\46&183\end{bmatrix}$, $\begin{bmatrix}217&16\\254&231\end{bmatrix}$, $\begin{bmatrix}229&100\\96&119\end{bmatrix}$, $\begin{bmatrix}279&232\\132&67\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.0.u.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 7 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^8\cdot7^4}\cdot\frac{(2x+y)^{48}(23750311936x^{16}+439667392512x^{15}y+3613787029504x^{14}y^{2}+17439258181632x^{13}y^{3}+55908421976064x^{12}y^{4}+127040888733696x^{11}y^{5}+211887120392192x^{10}y^{6}+264093484425216x^{9}y^{7}+247602642257408x^{8}y^{8}+173888969926656x^{7}y^{9}+89664455111168x^{6}y^{10}+32252814429696x^{5}y^{11}+7115392127424x^{4}y^{12}+619826481792x^{3}y^{13}-19552871456x^{2}y^{14}+32189964000xy^{15}+11296844833y^{16})^{3}}{(2x+y)^{48}(2x^{2}+3xy+2y^{2})^{4}(4x^{2}-4xy-7y^{2})^{8}(4x^{2}+20xy-3y^{2})^{2}(20x^{2}+44xy+13y^{2})^{8}(36x^{2}+68xy+29y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.48.0-8.e.1.5 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 280.48.0-8.e.1.14 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.h.1.18 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.h.1.27 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.m.1.8 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.m.1.15 | $280$ | $2$ | $2$ | $0$ | $?$ |
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.h.2.5 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.i.1.4 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.x.1.2 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.y.1.6 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.bk.2.3 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.bl.1.2 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.bo.1.1 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.bp.1.4 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.iy.2.15 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.iz.2.11 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.je.1.9 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.jf.1.11 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.ke.1.15 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.kf.1.13 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.kk.2.3 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.kl.2.11 | $280$ | $2$ | $2$ | $1$ |
| 280.480.16-280.df.2.8 | $280$ | $5$ | $5$ | $16$ |