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 | $2^{4}\cdot4^{2}\cdot8^{4}$ | 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: | 8O0 |
Level structure
| $\GL_2(\Z/280\Z)$-generators: | $\begin{bmatrix}17&196\\136&31\end{bmatrix}$, $\begin{bmatrix}151&216\\266&209\end{bmatrix}$, $\begin{bmatrix}181&260\\126&97\end{bmatrix}$, $\begin{bmatrix}241&76\\46&7\end{bmatrix}$, $\begin{bmatrix}249&240\\194&271\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.0.q.2 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 3 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4\cdot3^8\cdot7^4}\cdot\frac{(x+7y)^{48}(8265121x^{16}+406987840x^{15}y+7317439248x^{14}y^{2}-8073788800x^{13}y^{3}+224308799792x^{12}y^{4}-2333026510080x^{11}y^{5}+18336966592x^{10}y^{6}+26608427507200x^{9}y^{7}-47952317538720x^{8}y^{8}-372517985100800x^{7}y^{9}+3594045452032x^{6}y^{10}+6401824743659520x^{5}y^{11}+8617046852809472x^{4}y^{12}+4342277387571200x^{3}y^{13}+55096922245628928x^{2}y^{14}-42902014299791360xy^{15}+12197575118315776y^{16})^{3}}{(x-2y)^{8}(x+7y)^{56}(x^{2}+14y^{2})^{4}(5x^{2}-56xy-70y^{2})^{8}(31x^{4}+1120x^{3}y-2604x^{2}y^{2}-15680xy^{3}+6076y^{4})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.48.0-8.e.2.1 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 280.48.0-8.e.2.8 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.h.1.12 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.h.1.13 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.l.1.4 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.l.1.12 | $280$ | $2$ | $2$ | $0$ | $?$ |
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.g.1.6 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.j.1.4 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.w.2.4 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.z.1.4 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.bc.2.4 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.bd.2.8 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.bg.1.6 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.bh.1.8 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.ii.2.14 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.ij.2.12 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.io.2.2 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.ip.2.2 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.jo.2.10 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.jp.2.4 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.ju.2.6 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.jv.2.10 | $280$ | $2$ | $2$ | $1$ |
| 280.480.16-280.cx.1.25 | $280$ | $5$ | $5$ | $16$ |