| Curve name |
$X_{28}$ |
| Index |
$12$ |
| 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 & 2 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 4 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 6 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{11}$ |
| Curves that $X_{28}$ minimally covers |
$X_{11}$ |
| Curves that minimally cover $X_{28}$ |
$X_{67}$, $X_{68}$, $X_{81}$, $X_{90}$, $X_{128}$, $X_{146}$ |
| Curves that minimally cover $X_{28}$ and have infinitely many rational
points. |
$X_{67}$, $X_{68}$, $X_{81}$, $X_{90}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{28}) = \mathbb{Q}(f_{28}), f_{11} =
-2f_{28}^{2} - 8\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 2703x - 54090$, with conductor $468$ |
| Generic density of odd order reductions |
$89/336$ |