| Curve name |
$X_{89}$ |
| Index |
$24$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 1 \\ 6 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 6 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 6 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{30}$ |
| Curves that $X_{89}$ minimally covers |
$X_{30}$ |
| Curves that minimally cover $X_{89}$ |
$X_{245}$, $X_{265}$, $X_{292}$, $X_{294}$, $X_{297}$, $X_{299}$ |
| Curves that minimally cover $X_{89}$ and have infinitely many rational
points. |
$X_{297}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{89}) = \mathbb{Q}(f_{89}), f_{30} =
\frac{8f_{89} - 8}{f_{89}^{2} - 2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 5620x + 160304$, with conductor $4056$ |
| Generic density of odd order reductions |
$137/448$ |