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: | $\begin{bmatrix}32&43\\29&22\end{bmatrix}$, $\begin{bmatrix}34&69\\53&34\end{bmatrix}$, $\begin{bmatrix}39&34\\18&39\end{bmatrix}$, $\begin{bmatrix}64&7\\49&6\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 80.48.0.bl.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 |
---|---|---|---|---|---|
16.48.0-16.e.1.15 | $16$ | $2$ | $2$ | $0$ | $0$ |
40.48.0-40.cb.2.10 | $40$ | $2$ | $2$ | $0$ | $0$ |
80.48.0-16.e.1.8 | $80$ | $2$ | $2$ | $0$ | $?$ |
80.48.0-80.o.1.29 | $80$ | $2$ | $2$ | $0$ | $?$ |
80.48.0-80.o.1.30 | $80$ | $2$ | $2$ | $0$ | $?$ |
80.48.0-40.cb.2.6 | $80$ | $2$ | $2$ | $0$ | $?$ |
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.r.2.9 | $80$ | $2$ | $2$ | $1$ |
80.192.1-80.u.1.5 | $80$ | $2$ | $2$ | $1$ |
80.192.1-80.bg.2.6 | $80$ | $2$ | $2$ | $1$ |
80.192.1-80.bw.2.6 | $80$ | $2$ | $2$ | $1$ |
80.192.1-80.ci.2.7 | $80$ | $2$ | $2$ | $1$ |
80.192.1-80.cn.2.7 | $80$ | $2$ | $2$ | $1$ |
80.192.1-80.cz.2.7 | $80$ | $2$ | $2$ | $1$ |
80.192.1-80.dc.2.3 | $80$ | $2$ | $2$ | $1$ |
80.480.16-80.cf.1.15 | $80$ | $5$ | $5$ | $16$ |
240.192.1-240.qi.2.9 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.qy.2.15 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.ro.2.11 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.se.2.15 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.st.2.15 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.tk.2.9 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.ua.2.14 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.up.2.11 | $240$ | $2$ | $2$ | $1$ |
240.288.8-240.vb.1.58 | $240$ | $3$ | $3$ | $8$ |
240.384.7-240.zw.1.63 | $240$ | $4$ | $4$ | $7$ |