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 | $4^{8}\cdot8^{2}$ | 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: | 8N0 |
Level structure
| $\GL_2(\Z/280\Z)$-generators: | $\begin{bmatrix}23&240\\96&133\end{bmatrix}$, $\begin{bmatrix}111&180\\172&69\end{bmatrix}$, $\begin{bmatrix}161&240\\20&231\end{bmatrix}$, $\begin{bmatrix}185&236\\88&75\end{bmatrix}$, $\begin{bmatrix}255&158\\268&49\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.0.j.1 for the level structure with $-I$) |
| Cyclic 280-isogeny field degree: | $96$ |
| 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 7 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^4\cdot7^2}\cdot\frac{(7x-y)^{48}(23249442433x^{16}+49715643824x^{15}y+50558951856x^{14}y^{2}+29146833856x^{13}y^{3}+18289242944x^{12}y^{4}-1581202560x^{11}y^{5}+4344542272x^{10}y^{6}+996050048x^{9}y^{7}-1306220832x^{8}y^{8}-284585728x^{7}y^{9}+354656512x^{6}y^{10}+36879360x^{5}y^{11}+121877504x^{4}y^{12}-55494656x^{3}y^{13}+27503616x^{2}y^{14}-7727104xy^{15}+1032448y^{16})^{3}}{(7x-y)^{48}(7x^{2}+2y^{2})^{4}(7x^{2}-14xy-2y^{2})^{8}(7x^{2}+4xy-2y^{2})^{4}(7x^{2}+7xy+4y^{2})^{4}(14x^{2}-7xy+2y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.48.0-8.e.2.16 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 280.48.0-28.c.1.2 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-28.c.1.3 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-8.e.2.2 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.i.1.12 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.i.1.22 | $280$ | $2$ | $2$ | $0$ | $?$ |
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.t.1.5 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.y.2.3 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.be.2.3 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.bg.1.5 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.bw.2.7 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.by.1.6 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.cc.1.7 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.cd.2.4 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.gb.2.15 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.gf.2.13 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.hg.2.13 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.hk.2.15 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.lo.2.13 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.ls.2.14 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.mu.2.14 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.my.2.13 | $280$ | $2$ | $2$ | $1$ |
| 280.480.16-280.bd.1.5 | $280$ | $5$ | $5$ | $16$ |