Invariants
| Level: | $280$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot4$ | ||
| 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: | 8G0 |
Level structure
| $\GL_2(\Z/280\Z)$-generators: | $\begin{bmatrix}49&246\\186&203\end{bmatrix}$, $\begin{bmatrix}101&156\\162&249\end{bmatrix}$, $\begin{bmatrix}107&94\\196&33\end{bmatrix}$, $\begin{bmatrix}229&186\\98&39\end{bmatrix}$, $\begin{bmatrix}237&182\\124&251\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.24.0.k.1 for the level structure with $-I$) |
| Cyclic 280-isogeny field degree: | $192$ |
| Cyclic 280-torsion field degree: | $18432$ |
| Full 280-torsion field degree: | $30965760$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 19 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4\cdot3^3}{5^4}\cdot\frac{x^{24}(27x^{4}+180x^{3}y+480x^{2}y^{2}+600xy^{3}+325y^{4})^{3}(81x^{4}+540x^{3}y+1260x^{2}y^{2}+1200xy^{3}+475y^{4})^{3}}{y^{8}x^{24}(3x+5y)^{8}(81x^{4}+540x^{3}y+1350x^{2}y^{2}+1500xy^{3}+725y^{4})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 28.24.0-4.a.1.2 | $28$ | $2$ | $2$ | $0$ | $0$ |
| 280.24.0-4.a.1.6 | $280$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 280.96.1-40.b.1.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-40.d.1.3 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-40.e.1.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-40.h.1.4 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-40.j.1.4 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-40.l.1.4 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-40.o.1.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-40.z.1.3 | $280$ | $2$ | $2$ | $1$ |
| 280.240.8-40.p.1.5 | $280$ | $5$ | $5$ | $8$ |
| 280.288.7-40.bl.1.9 | $280$ | $6$ | $6$ | $7$ |
| 280.480.15-40.cb.1.13 | $280$ | $10$ | $10$ | $15$ |
| 280.96.1-280.eg.1.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.eh.1.6 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.ek.1.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.el.1.6 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.eo.1.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.ep.1.6 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.es.1.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.et.1.6 | $280$ | $2$ | $2$ | $1$ |
| 280.384.11-280.cz.1.25 | $280$ | $8$ | $8$ | $11$ |