| Curve name |
$X_{81}$ |
| Index |
$24$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 6 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 2 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{28}$ |
| Curves that $X_{81}$ minimally covers |
$X_{28}$, $X_{41}$, $X_{45}$ |
| Curves that minimally cover $X_{81}$ |
$X_{218}$, $X_{220}$, $X_{262}$, $X_{264}$, $X_{363}$, $X_{364}$ |
| Curves that minimally cover $X_{81}$ and have infinitely many rational
points. |
$X_{218}$, $X_{220}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{81}) = \mathbb{Q}(f_{81}), f_{28} =
\frac{8f_{81}}{f_{81}^{2} - 2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 129800x + 17843712$, with conductor $202496$ |
| Generic density of odd order reductions |
$1427/5376$ |