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: | $10$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8O0 |
Level structure
| $\GL_2(\Z/80\Z)$-generators: | $\begin{bmatrix}21&48\\33&21\end{bmatrix}$, $\begin{bmatrix}39&72\\56&57\end{bmatrix}$, $\begin{bmatrix}43&72\\5&7\end{bmatrix}$, $\begin{bmatrix}63&40\\26&61\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.48.0.bp.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
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 5 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2\cdot5}\cdot\frac{(x-y)^{48}(390625x^{16}+36250000x^{14}y^{2}+45750000x^{12}y^{4}-29800000x^{10}y^{6}+7100000x^{8}y^{8}-4768000x^{6}y^{10}+1171200x^{4}y^{12}+148480x^{2}y^{14}+256y^{16})^{3}}{y^{2}x^{2}(x-y)^{48}(5x^{2}-2y^{2})^{2}(5x^{2}+2y^{2})^{4}(25x^{4}-60x^{2}y^{2}+4y^{4})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.48.0-8.bb.2.7 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 80.48.0-8.bb.2.5 | $80$ | $2$ | $2$ | $0$ | $?$ |
| 80.48.0-40.bp.1.3 | $80$ | $2$ | $2$ | $0$ | $?$ |
| 80.48.0-40.bp.1.7 | $80$ | $2$ | $2$ | $0$ | $?$ |
| 80.48.0-40.cb.2.8 | $80$ | $2$ | $2$ | $0$ | $?$ |
| 80.48.0-40.cb.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.cz.2.3 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.db.1.3 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.dh.1.3 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.dj.1.2 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.eh.1.5 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.ej.2.2 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.ep.2.3 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.er.2.3 | $80$ | $2$ | $2$ | $1$ |
| 80.480.16-40.cf.2.8 | $80$ | $5$ | $5$ | $16$ |
| 240.192.1-240.pm.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.po.1.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.qc.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.qe.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.yc.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ye.2.2 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ys.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.yu.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.288.8-120.sh.1.14 | $240$ | $3$ | $3$ | $8$ |
| 240.384.7-120.ma.1.13 | $240$ | $4$ | $4$ | $7$ |