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 | $4^{8}\cdot8^{2}$ | 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: | 8N0 |
Level structure
| $\GL_2(\Z/280\Z)$-generators: | $\begin{bmatrix}155&216\\268&239\end{bmatrix}$, $\begin{bmatrix}177&178\\272&45\end{bmatrix}$, $\begin{bmatrix}197&128\\116&161\end{bmatrix}$, $\begin{bmatrix}217&128\\76&177\end{bmatrix}$, $\begin{bmatrix}273&142\\4&43\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.0.m.2 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 3 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^2\cdot7^2\cdot13^4}\cdot\frac{(13x+21y)^{48}(25062997100294179876025761x^{16}-307705846736097975227273120x^{15}y+1965718506844215013023679728x^{14}y^{2}-4583890911009010046502521920x^{13}y^{3}-2865143833263852701323589008x^{12}y^{4}+4466054984127655662764791680x^{11}y^{5}+132433219475516329981626713152x^{10}y^{6}-431217221352070840571468729600x^{9}y^{7}+449894942809238010488664871520x^{8}y^{8}+4225756778931681540432704000x^{7}y^{9}-229213437926306425204119107328x^{6}y^{10}-70693897898348769369349606400x^{5}y^{11}+275105613984775507095512127232x^{4}y^{12}-166944756074935774493512980480x^{3}y^{13}+46043711553221716638846983168x^{2}y^{14}-9539617665203183675118776320xy^{15}+1981794804690161387706040576y^{16})^{3}}{(13x-16y)^{4}(13x+21y)^{52}(65x^{2}-308xy+70y^{2})^{4}(1521x^{2}-1820xy+2674y^{2})^{8}(885391x^{4}+7997080x^{3}y-23693124x^{2}y^{2}+4841200xy^{3}+5494076y^{4})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.48.0-8.e.1.9 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 280.48.0-8.e.1.14 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.e.1.1 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.e.1.4 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.h.2.2 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.h.2.25 | $280$ | $2$ | $2$ | $0$ | $?$ |
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.j.2.2 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.z.2.5 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.bk.2.3 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.bo.2.2 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.bv.1.5 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.bz.2.2 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.cf.2.2 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.ch.1.3 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.gn.1.11 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.gt.1.14 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.hs.1.14 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.hy.1.9 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.mb.1.12 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.mh.1.13 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.nh.1.13 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.nn.1.10 | $280$ | $2$ | $2$ | $1$ |
| 280.480.16-280.bj.2.7 | $280$ | $5$ | $5$ | $16$ |