| Curve name |
$X_{97}$ |
| Index |
$24$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 3 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 2 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 6 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{39}$ |
| Curves that $X_{97}$ minimally covers |
$X_{39}$, $X_{45}$, $X_{47}$ |
| Curves that minimally cover $X_{97}$ |
$X_{254}$, $X_{260}$, $X_{264}$, $X_{275}$, $X_{301}$, $X_{303}$, $X_{304}$, $X_{308}$, $X_{309}$, $X_{310}$, $X_{311}$, $X_{312}$, $X_{377}$, $X_{380}$ |
| Curves that minimally cover $X_{97}$ and have infinitely many rational
points. |
$X_{304}$, $X_{308}$, $X_{309}$, $X_{312}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{97}) = \mathbb{Q}(f_{97}), f_{39} =
\frac{4f_{97}^{2} + 1}{f_{97}^{2} + f_{97} - \frac{1}{4}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 558481500x - 5079892400000$, with conductor $1159200$ |
| Generic density of odd order reductions |
$1343/5376$ |