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: | $\begin{bmatrix}4&59\\25&78\end{bmatrix}$, $\begin{bmatrix}24&9\\3&62\end{bmatrix}$, $\begin{bmatrix}48&65\\31&18\end{bmatrix}$, $\begin{bmatrix}62&1\\77&34\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 80.48.0.bn.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.1.8 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 80.48.0-80.m.1.17 | $80$ | $2$ | $2$ | $0$ | $?$ |
| 80.48.0-80.m.1.24 | $80$ | $2$ | $2$ | $0$ | $?$ |
| 80.48.0-80.o.1.18 | $80$ | $2$ | $2$ | $0$ | $?$ |
| 80.48.0-80.o.1.19 | $80$ | $2$ | $2$ | $0$ | $?$ |
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.2.1 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.v.2.1 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.bj.2.7 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.bu.2.3 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.cj.1.3 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.cp.1.5 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.db.2.2 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.dd.2.7 | $80$ | $2$ | $2$ | $1$ |
| 80.480.16-80.ch.1.9 | $80$ | $5$ | $5$ | $16$ |
| 240.192.1-240.qk.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ra.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.rq.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.sg.2.2 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.sw.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.to.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ue.2.2 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.us.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.288.8-240.vf.1.19 | $240$ | $3$ | $3$ | $8$ |
| 240.384.7-240.baa.1.33 | $240$ | $4$ | $4$ | $7$ |