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: | $10$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $2^{5}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8O0 |
Level structure
$\GL_2(\Z/80\Z)$-generators: | $\begin{bmatrix}3&8\\40&51\end{bmatrix}$, $\begin{bmatrix}19&24\\0&31\end{bmatrix}$, $\begin{bmatrix}39&8\\16&75\end{bmatrix}$, $\begin{bmatrix}55&72\\22&39\end{bmatrix}$, $\begin{bmatrix}59&48\\70&49\end{bmatrix}$, $\begin{bmatrix}65&32\\28&49\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 40.48.0.bd.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 3 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^2\cdot5^2}\cdot\frac{(4x+y)^{48}(4043309056x^{16}+43956305920x^{15}y+182452224000x^{14}y^{2}+348127232000x^{13}y^{3}+401997824000x^{12}y^{4}+149422080000x^{11}y^{5}-152453120000x^{10}y^{6}-167567360000x^{9}y^{7}-97896960000x^{8}y^{8}-104729600000x^{7}y^{9}-59552000000x^{6}y^{10}+36480000000x^{5}y^{11}+61340000000x^{4}y^{12}+33200000000x^{3}y^{13}+10875000000x^{2}y^{14}+1637500000xy^{15}+94140625y^{16})^{3}}{(4x+y)^{48}(4x^{2}-10xy-5y^{2})^{8}(8x^{2}-5y^{2})^{2}(8x^{2}+4xy+5y^{2})^{4}(8x^{2}+40xy+5y^{2})^{2}(16x^{2}+20xy-5y^{2})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
16.48.0-8.i.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ |
80.48.0-8.i.1.4 | $80$ | $2$ | $2$ | $0$ | $?$ |
80.48.0-40.cb.2.1 | $80$ | $2$ | $2$ | $0$ | $?$ |
80.48.0-40.cb.2.5 | $80$ | $2$ | $2$ | $0$ | $?$ |
80.48.0-40.cb.2.12 | $80$ | $2$ | $2$ | $0$ | $?$ |
80.48.0-40.cb.2.16 | $80$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.