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 | $2^{3}\cdot4$ | ||
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: | $\begin{bmatrix}3&16\\79&57\end{bmatrix}$, $\begin{bmatrix}43&48\\78&21\end{bmatrix}$, $\begin{bmatrix}73&64\\26&61\end{bmatrix}$, $\begin{bmatrix}77&24\\8&19\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 40.48.0.bi.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 $ | $=$ | $ 3 x^{2} + 4 x z + 40 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.ba.2.8 | $16$ | $2$ | $2$ | $0$ | $0$ |
80.48.0-8.ba.2.1 | $80$ | $2$ | $2$ | $0$ | $?$ |
80.48.0-40.bj.1.5 | $80$ | $2$ | $2$ | $0$ | $?$ |
80.48.0-40.bj.1.7 | $80$ | $2$ | $2$ | $0$ | $?$ |
80.48.0-40.ca.2.2 | $80$ | $2$ | $2$ | $0$ | $?$ |
80.48.0-40.ca.2.12 | $80$ | $2$ | $2$ | $0$ | $?$ |
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.ce.2.1 | $80$ | $2$ | $2$ | $1$ |
80.192.1-80.cg.2.1 | $80$ | $2$ | $2$ | $1$ |
80.192.1-80.cm.1.1 | $80$ | $2$ | $2$ | $1$ |
80.192.1-80.co.1.1 | $80$ | $2$ | $2$ | $1$ |
80.192.1-80.dm.1.1 | $80$ | $2$ | $2$ | $1$ |
80.192.1-80.do.1.1 | $80$ | $2$ | $2$ | $1$ |
80.192.1-80.du.2.1 | $80$ | $2$ | $2$ | $1$ |
80.192.1-80.dw.2.1 | $80$ | $2$ | $2$ | $1$ |
80.480.16-40.by.1.4 | $80$ | $5$ | $5$ | $16$ |
240.192.1-240.nx.2.5 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.nz.2.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.on.2.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.op.2.3 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.wn.2.3 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.wp.2.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.xd.2.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.xf.2.5 | $240$ | $2$ | $2$ | $1$ |
240.288.8-120.rc.1.10 | $240$ | $3$ | $3$ | $8$ |
240.384.7-120.lf.1.26 | $240$ | $4$ | $4$ | $7$ |