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 | $1^{4}\cdot2^{2}\cdot4^{2}\cdot16^{2}$ | 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: | 16H0 |
Level structure
| $\GL_2(\Z/80\Z)$-generators: | $\begin{bmatrix}3&4\\6&17\end{bmatrix}$, $\begin{bmatrix}14&19\\17&48\end{bmatrix}$, $\begin{bmatrix}15&26\\34&7\end{bmatrix}$, $\begin{bmatrix}44&19\\37&34\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 80.48.0.bp.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 but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
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 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.bb.1.7 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 80.48.0-80.n.2.1 | $80$ | $2$ | $2$ | $0$ | $?$ |
| 80.48.0-80.n.2.16 | $80$ | $2$ | $2$ | $0$ | $?$ |
| 80.48.0-80.o.1.2 | $80$ | $2$ | $2$ | $0$ | $?$ |
| 80.48.0-80.o.1.30 | $80$ | $2$ | $2$ | $0$ | $?$ |
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.i.2.9 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.w.2.3 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.bn.1.10 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.bu.1.3 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.cl.2.2 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.cn.2.7 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.cz.1.3 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.df.1.5 | $80$ | $2$ | $2$ | $1$ |
| 80.480.16-80.cj.1.3 | $80$ | $5$ | $5$ | $16$ |
| 240.192.1-240.qm.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.rc.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.rs.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.si.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.sy.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.tm.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.uc.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.uu.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.288.8-240.wb.1.7 | $240$ | $3$ | $3$ | $8$ |
| 240.384.7-240.bag.1.3 | $240$ | $4$ | $4$ | $7$ |