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: | $2$ | ||||||
$\overline{\Q}$-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: | $\begin{bmatrix}1&40\\38&47\end{bmatrix}$, $\begin{bmatrix}45&8\\62&3\end{bmatrix}$, $\begin{bmatrix}51&40\\12&39\end{bmatrix}$, $\begin{bmatrix}59&48\\72&17\end{bmatrix}$, $\begin{bmatrix}65&16\\58&53\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 40.48.0.bb.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 $ | $=$ | $ 5 x^{2} + 5 y^{2} + 8 z^{2} $ |
Rational points
This modular curve has no real points, and therefore no 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.i.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ |
80.48.0-8.i.1.4 | $80$ | $2$ | $2$ | $0$ | $?$ |
80.48.0-40.ca.1.1 | $80$ | $2$ | $2$ | $0$ | $?$ |
80.48.0-40.ca.1.8 | $80$ | $2$ | $2$ | $0$ | $?$ |
80.48.0-40.ca.1.9 | $80$ | $2$ | $2$ | $0$ | $?$ |
80.48.0-40.ca.1.16 | $80$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.