| Curve name |
$X_{73}$ |
| Index |
$24$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 5 \\ 4 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 4 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{26}$ |
| Curves that $X_{73}$ minimally covers |
$X_{26}$, $X_{45}$, $X_{50}$ |
| Curves that minimally cover $X_{73}$ |
$X_{253}$, $X_{267}$, $X_{276}$, $X_{277}$, $X_{348}$, $X_{349}$, $X_{350}$, $X_{351}$ |
| Curves that minimally cover $X_{73}$ and have infinitely many rational
points. |
$X_{349}$, $X_{350}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{73}) = \mathbb{Q}(f_{73}), f_{26} =
\frac{f_{73}^{2} - 8}{f_{73}^{2} + 8f_{73} + 8}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + 787952x + 97113753$, with conductor $16940$ |
| Generic density of odd order reductions |
$1427/5376$ |