Invariants
| Level: | $80$ | $\SL_2$-level: | $16$ | ||||
| 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 | $1^{2}\cdot2\cdot4\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| 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: | 8I0 |
Level structure
| $\GL_2(\Z/80\Z)$-generators: | $\begin{bmatrix}3&20\\48&63\end{bmatrix}$, $\begin{bmatrix}25&28\\38&15\end{bmatrix}$, $\begin{bmatrix}59&28\\74&21\end{bmatrix}$, $\begin{bmatrix}78&33\\59&76\end{bmatrix}$, $\begin{bmatrix}78&75\\49&8\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.24.0.ca.1 for the level structure with $-I$) |
| Cyclic 80-isogeny field degree: | $12$ |
| Cyclic 80-torsion field degree: | $192$ |
| Full 80-torsion field degree: | $245760$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 55 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^4}{5^2}\cdot\frac{x^{24}(625x^{8}-3500x^{6}y^{2}-250x^{4}y^{4}+20x^{2}y^{6}+y^{8})^{3}}{y^{2}x^{28}(5x^{2}+y^{2})^{8}(10x^{2}+y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.24.0-8.n.1.8 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 80.24.0-8.n.1.4 | $80$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.