| Curve name |
$X_{69}$ |
| Index |
$24$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 2 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 2 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 4 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 6 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{23}$ |
| Curves that $X_{69}$ minimally covers |
$X_{23}$, $X_{45}$, $X_{50}$ |
| Curves that minimally cover $X_{69}$ |
$X_{250}$, $X_{251}$, $X_{252}$, $X_{258}$, $X_{262}$, $X_{277}$, $X_{319}$, $X_{320}$, $X_{321}$, $X_{322}$, $X_{323}$, $X_{324}$ |
| Curves that minimally cover $X_{69}$ and have infinitely many rational
points. |
$X_{320}$, $X_{323}$, $X_{324}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{69}) = \mathbb{Q}(f_{69}), f_{23} =
\frac{f_{69}^{2} + 2}{f_{69}^{2} - 2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 + 529x - 20491$, with conductor $11109$ |
| Generic density of odd order reductions |
$401/1792$ |