| Curve name |
$X_{68}$ |
| Index |
$24$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 2 & 3 \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 $X_{68}$ minimally covers |
$X_{23}$, $X_{28}$, $X_{47}$ |
| Curves that minimally cover $X_{68}$ |
$X_{255}$, $X_{264}$ |
| Curves that minimally cover $X_{68}$ and have infinitely many rational
points. |
|
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{68}) = \mathbb{Q}(f_{68}), f_{23} =
\frac{f_{68}}{f_{68}^{2} + \frac{1}{4}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 805514x - 264095580$, with conductor $16810$ |
| Generic density of odd order reductions |
$109/448$ |