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: | $\begin{bmatrix}23&24\\3&63\end{bmatrix}$, $\begin{bmatrix}31&56\\58&7\end{bmatrix}$, $\begin{bmatrix}41&0\\12&11\end{bmatrix}$, $\begin{bmatrix}71&8\\5&3\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.48.0.bj.1 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 6 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^2\cdot3^4\cdot5}\cdot\frac{(4x-3y)^{48}(1739478532096x^{16}-6089525428224x^{15}y+5589832826880x^{14}y^{2}+3833384140800x^{13}y^{3}-11427533291520x^{12}y^{4}+9071276064768x^{11}y^{5}-1634312650752x^{10}y^{6}-4653835223040x^{9}y^{7}+10570831257600x^{8}y^{8}-12268990218240x^{7}y^{9}+8419794075648x^{6}y^{10}-3663161874432x^{5}y^{11}+1068664078080x^{4}y^{12}-214277011200x^{3}y^{13}+24488801280x^{2}y^{14}-459165024xy^{15}+43046721y^{16})^{3}}{x^{2}(2x-3y)^{2}(4x-3y)^{48}(8x^{2}-4xy+3y^{2})^{4}(16x^{2}+12xy-9y^{2})^{2}(64x^{4}-1344x^{3}y+864x^{2}y^{2}+216xy^{3}-81y^{4})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.48.0-8.bb.2.8 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 80.48.0-8.bb.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.cb.1.2 | $80$ | $2$ | $2$ | $0$ | $?$ |
| 80.48.0-40.cb.1.8 | $80$ | $2$ | $2$ | $0$ | $?$ |
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.cf.2.1 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.ch.2.1 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.cn.1.3 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.cp.2.1 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.dn.1.2 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.dp.1.1 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.dv.2.1 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.dx.1.1 | $80$ | $2$ | $2$ | $1$ |
| 80.480.16-40.bz.2.6 | $80$ | $5$ | $5$ | $16$ |
| 240.192.1-240.ny.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.oa.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.oo.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.oq.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.wo.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.wq.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.xe.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.xg.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.288.8-120.rj.2.20 | $240$ | $3$ | $3$ | $8$ |
| 240.384.7-120.li.2.10 | $240$ | $4$ | $4$ | $7$ |