| Curve name |
$X_{209}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 2 \\ 14 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 10 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{61}$ |
| Curves that $X_{209}$ minimally covers |
$X_{61}$, $X_{109}$, $X_{110}$ |
| Curves that minimally cover $X_{209}$ |
$X_{209a}$, $X_{209b}$, $X_{209c}$, $X_{209d}$ |
| Curves that minimally cover $X_{209}$ and have infinitely many rational
points. |
$X_{209a}$, $X_{209b}$, $X_{209c}$, $X_{209d}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{209}) = \mathbb{Q}(f_{209}), f_{61} =
-f_{209}^{2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 12336x + 530432$, with conductor $4626$ |
| Generic density of odd order reductions |
$9249/57344$ |