Invariants
| Level: | $80$ | $\SL_2$-level: | $16$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $2^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8O0 |
Level structure
| $\GL_2(\Z/80\Z)$-generators: | $\begin{bmatrix}5&48\\76&47\end{bmatrix}$, $\begin{bmatrix}37&16\\35&49\end{bmatrix}$, $\begin{bmatrix}63&24\\32&25\end{bmatrix}$, $\begin{bmatrix}63&24\\73&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.48.0.bn.2 for the level structure with $-I$) |
| Cyclic 80-isogeny field degree: | $12$ |
| Cyclic 80-torsion field degree: | $192$ |
| Full 80-torsion field degree: | $122880$ |
Models
Smooth plane model Smooth plane model
| $ 0 $ | $=$ | $ 10 x^{2} + 4 y^{2} + 2 y z - z^{2} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.48.0-8.bb.1.6 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 80.48.0-8.bb.1.1 | $80$ | $2$ | $2$ | $0$ | $?$ |
| 80.48.0-40.bn.1.6 | $80$ | $2$ | $2$ | $0$ | $?$ |
| 80.48.0-40.bn.1.7 | $80$ | $2$ | $2$ | $0$ | $?$ |
| 80.48.0-40.ca.1.8 | $80$ | $2$ | $2$ | $0$ | $?$ |
| 80.48.0-40.ca.1.10 | $80$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 80.192.1-80.cv.1.5 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.cx.1.2 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.dd.2.7 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.df.1.5 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.ed.2.6 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.ef.2.3 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.el.1.2 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.en.2.2 | $80$ | $2$ | $2$ | $1$ |
| 80.480.16-40.cd.1.4 | $80$ | $5$ | $5$ | $16$ |
| 240.192.1-240.pe.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.pg.2.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.pu.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.pw.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.xu.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.xw.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.yk.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ym.2.3 | $240$ | $2$ | $2$ | $1$ |
| 240.288.8-120.rz.1.22 | $240$ | $3$ | $3$ | $8$ |
| 240.384.7-120.lu.1.14 | $240$ | $4$ | $4$ | $7$ |