Invariants
| Level: | $80$ | $\SL_2$-level: | $80$ | Newform level: | $1$ | ||
| Index: | $480$ | $\PSL_2$-index: | $240$ | ||||
| Genus: | $16 = 1 + \frac{ 240 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (none of which are rational) | Cusp widths | $5^{4}\cdot10^{2}\cdot20^{2}\cdot80^{2}$ | Cusp orbits | $2^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 30$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 16$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 80B16 |
Level structure
| $\GL_2(\Z/80\Z)$-generators: | $\begin{bmatrix}42&73\\73&58\end{bmatrix}$, $\begin{bmatrix}48&9\\67&22\end{bmatrix}$, $\begin{bmatrix}50&31\\9&56\end{bmatrix}$, $\begin{bmatrix}67&78\\22&75\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 80.240.16.da.2 for the level structure with $-I$) |
| Cyclic 80-isogeny field degree: | $6$ |
| Cyclic 80-torsion field degree: | $96$ |
| Full 80-torsion field degree: | $24576$ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=31$, and therefore no 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 |
|---|---|---|---|---|---|
| $X_{S_4}(5)$ | $5$ | $96$ | $48$ | $0$ | $0$ |
| 16.96.0-16.w.1.1 | $16$ | $5$ | $5$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.96.0-16.w.1.1 | $16$ | $5$ | $5$ | $0$ | $0$ |
| 40.240.8-40.da.2.7 | $40$ | $2$ | $2$ | $8$ | $2$ |
| 80.240.8-80.t.2.7 | $80$ | $2$ | $2$ | $8$ | $?$ |
| 80.240.8-80.t.2.9 | $80$ | $2$ | $2$ | $8$ | $?$ |
| 80.240.8-80.y.1.1 | $80$ | $2$ | $2$ | $8$ | $?$ |
| 80.240.8-80.y.1.19 | $80$ | $2$ | $2$ | $8$ | $?$ |
| 80.240.8-40.da.2.1 | $80$ | $2$ | $2$ | $8$ | $?$ |