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: | $\begin{bmatrix}17&160\\192&191\end{bmatrix}$, $\begin{bmatrix}167&248\\160&71\end{bmatrix}$, $\begin{bmatrix}251&136\\130&39\end{bmatrix}$, $\begin{bmatrix}279&136\\177&257\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.0.bn.1 for the level structure with $-I$) |
| Cyclic 280-isogeny field degree: | $48$ |
| Cyclic 280-torsion field degree: | $2304$ |
| 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2\cdot7}\cdot\frac{(x+2y)^{48}(47041x^{16}-1128736x^{15}y-6802208x^{14}y^{2}-169714048x^{13}y^{3}+865658304x^{12}y^{4}+16926748160x^{11}y^{5}-15121415680x^{10}y^{6}+597332535296x^{9}y^{7}-3184996186624x^{8}y^{8}-16725310988288x^{7}y^{9}-11855189893120x^{6}y^{10}-371575975608320x^{5}y^{11}+532082070503424x^{4}y^{12}+2920841220849664x^{3}y^{13}-3277918080991232x^{2}y^{14}+15229954156920832xy^{15}+17772183803723776y^{16})^{3}}{(x+2y)^{48}(x^{2}+28y^{2})^{2}(x^{2}-4xy-28y^{2})^{2}(x^{2}+28xy-28y^{2})^{4}(3x^{2}-28xy+140y^{2})^{8}(5x^{2}+28xy+84y^{2})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.48.0-8.bb.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 280.48.0-8.bb.1.7 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.bl.1.4 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.bl.1.9 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.bv.2.1 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.bv.2.2 | $280$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 280.480.16-280.fj.2.13 | $280$ | $5$ | $5$ | $16$ |