| Curve name |
$X_{116}$ |
| Index |
$24$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 9 \\ 12 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{32}$ |
| Curves that $X_{116}$ minimally covers |
$X_{32}$ |
| Curves that minimally cover $X_{116}$ |
$X_{241}$, $X_{242}$, $X_{316}$, $X_{333}$, $X_{407}$, $X_{408}$, $X_{116a}$, $X_{116b}$, $X_{116c}$, $X_{116d}$, $X_{116e}$, $X_{116f}$, $X_{116g}$, $X_{116h}$, $X_{116i}$, $X_{116j}$, $X_{116k}$, $X_{116l}$ |
| Curves that minimally cover $X_{116}$ and have infinitely many rational
points. |
$X_{241}$, $X_{242}$, $X_{116a}$, $X_{116b}$, $X_{116c}$, $X_{116d}$, $X_{116e}$, $X_{116f}$, $X_{116g}$, $X_{116h}$, $X_{116i}$, $X_{116j}$, $X_{116k}$, $X_{116l}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{116}) = \mathbb{Q}(f_{116}), f_{32} =
-f_{116}^{2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - 4101x + 100723$, with conductor $7275$ |
| Generic density of odd order reductions |
$289/1792$ |