| Curve name |
$X_{109}$ |
| Index |
$24$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 14 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{45}$ |
| Curves that $X_{109}$ minimally covers |
$X_{45}$ |
| Curves that minimally cover $X_{109}$ |
$X_{209}$, $X_{220}$, $X_{311}$, $X_{320}$, $X_{382}$, $X_{401}$ |
| Curves that minimally cover $X_{109}$ and have infinitely many rational
points. |
$X_{209}$, $X_{220}$, $X_{320}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{109}) = \mathbb{Q}(f_{109}), f_{45} =
-2f_{109}^{2} - 4\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 2000x - 34375$, with conductor $700$ |
| Generic density of odd order reductions |
$85091/344064$ |