Invariants
| Level: | $80$ | $\SL_2$-level: | $80$ | Newform level: | $1$ | ||
| Index: | $240$ | $\PSL_2$-index: | $120$ | ||||
| Genus: | $9 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $10^{2}\cdot20\cdot80$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 9$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 9$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 80A9 |
Level structure
| $\GL_2(\Z/80\Z)$-generators: | $\begin{bmatrix}26&11\\45&4\end{bmatrix}$, $\begin{bmatrix}26&69\\21&50\end{bmatrix}$, $\begin{bmatrix}47&40\\62&69\end{bmatrix}$, $\begin{bmatrix}55&4\\18&5\end{bmatrix}$, $\begin{bmatrix}70&59\\73&60\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 80.120.9.b.1 for the level structure with $-I$) |
| Cyclic 80-isogeny field degree: | $12$ |
| Cyclic 80-torsion field degree: | $192$ |
| Full 80-torsion field degree: | $49152$ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_{S_4}(5)$ | $5$ | $48$ | $24$ | $0$ | $0$ |
| 16.48.1-16.b.1.6 | $16$ | $5$ | $5$ | $1$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.48.1-16.b.1.6 | $16$ | $5$ | $5$ | $1$ | $0$ |
| 40.120.4-40.bl.1.10 | $40$ | $2$ | $2$ | $4$ | $0$ |
| 80.120.4-40.bl.1.5 | $80$ | $2$ | $2$ | $4$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.