| Curve name |
$X_{65}$ |
| Index |
$24$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 5 \\ 2 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 7 \\ 2 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{23}$ |
| Curves that $X_{65}$ minimally covers |
$X_{23}$, $X_{35}$, $X_{50}$ |
| Curves that minimally cover $X_{65}$ |
$X_{251}$, $X_{261}$, $X_{317}$, $X_{318}$ |
| Curves that minimally cover $X_{65}$ and have infinitely many rational
points. |
$X_{318}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{65}) = \mathbb{Q}(f_{65}), f_{23} =
\frac{f_{65}^{2} - 2}{f_{65}^{2} + 2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 + x^2 - 23x + 20$, with conductor $867$ |
| Generic density of odd order reductions |
$401/1792$ |