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: | $\begin{bmatrix}45&186\\104&35\end{bmatrix}$, $\begin{bmatrix}79&218\\128&245\end{bmatrix}$, $\begin{bmatrix}119&240\\202&227\end{bmatrix}$, $\begin{bmatrix}205&86\\248&235\end{bmatrix}$, $\begin{bmatrix}223&50\\198&221\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.24.0.j.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 24 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^8\cdot5^4}\cdot\frac{x^{24}(25x^{4}-80x^{2}y^{2}+256y^{4})^{3}(25x^{4}+80x^{2}y^{2}+256y^{4})^{3}}{y^{8}x^{32}(25x^{4}+256y^{4})^{2}}$ |
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.1.5 | $280$ | $2$ | $2$ | $0$ | $?$ |
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.a.1.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-40.c.1.4 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-40.f.1.4 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-40.g.1.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-40.i.1.4 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-40.k.1.4 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-40.n.1.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-40.t.1.4 | $280$ | $2$ | $2$ | $1$ |
| 280.240.8-40.o.1.5 | $280$ | $5$ | $5$ | $8$ |
| 280.288.7-40.bk.1.9 | $280$ | $6$ | $6$ | $7$ |
| 280.480.15-40.ca.1.13 | $280$ | $10$ | $10$ | $15$ |
| 280.96.1-280.ee.1.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.ef.1.3 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.ei.1.3 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.ej.1.5 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.em.1.3 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.en.1.3 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.eq.1.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.er.1.3 | $280$ | $2$ | $2$ | $1$ |
| 280.384.11-280.cy.1.25 | $280$ | $8$ | $8$ | $11$ |