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: | $\begin{bmatrix}11&176\\20&143\end{bmatrix}$, $\begin{bmatrix}45&78\\212&147\end{bmatrix}$, $\begin{bmatrix}75&114\\108&5\end{bmatrix}$, $\begin{bmatrix}133&58\\44&261\end{bmatrix}$, $\begin{bmatrix}183&248\\124&73\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.0.n.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 2 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6\cdot3^4\cdot7^2}\cdot\frac{x^{48}(5764801x^{16}+474360768x^{14}y^{2}+153692888832x^{12}y^{4}+5972066537472x^{10}y^{6}+107755294187520x^{8}y^{8}+631820263882752x^{6}y^{10}+1720250130235392x^{4}y^{12}+561714328240128x^{2}y^{14}+722204136308736y^{16})^{3}}{y^{4}x^{52}(7x^{2}-72y^{2})^{8}(7x^{2}+72y^{2})^{4}(49x^{4}+3024x^{2}y^{2}+5184y^{4})^{4}}$ |
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.1.6 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.e.1.1 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.e.1.18 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.i.1.9 | $280$ | $2$ | $2$ | $0$ | $?$ |
| 280.48.0-56.i.1.29 | $280$ | $2$ | $2$ | $0$ | $?$ |
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.u.1.5 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.z.2.5 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.bm.1.5 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.bo.1.2 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.bx.1.5 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.bz.1.5 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.cg.1.5 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-56.ch.1.3 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.gr.2.11 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.gv.1.16 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.hw.1.14 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.ia.2.3 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.mf.1.16 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.mj.2.10 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.nl.2.2 | $280$ | $2$ | $2$ | $1$ |
| 280.192.1-280.np.1.12 | $280$ | $2$ | $2$ | $1$ |
| 280.480.16-280.bl.2.30 | $280$ | $5$ | $5$ | $16$ |