| Curve name |
$X_{24}$ |
| Index |
$12$ |
| Level |
$4$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 2 \\ 2 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
| Level | Index of image | Corresponding curve |
| $2$ |
$6$ |
$X_{8}$ |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{8}$ |
| Curves that $X_{24}$ minimally covers |
$X_{8}$, $X_{9}$, $X_{11}$ |
| Curves that minimally cover $X_{24}$ |
$X_{58}$, $X_{59}$, $X_{61}$, $X_{66}$, $X_{67}$, $X_{132}$, $X_{141}$, $X_{24a}$, $X_{24b}$, $X_{24c}$, $X_{24d}$, $X_{24e}$, $X_{24f}$, $X_{24g}$, $X_{24h}$ |
| Curves that minimally cover $X_{24}$ and have infinitely many rational
points. |
$X_{58}$, $X_{61}$, $X_{66}$, $X_{67}$, $X_{24a}$, $X_{24b}$, $X_{24c}$, $X_{24d}$, $X_{24e}$, $X_{24f}$, $X_{24g}$, $X_{24h}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{24}) = \mathbb{Q}(f_{24}), f_{8} =
-2f_{24}^{2} - 1\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 240x - 469$, with conductor $1845$ |
| Generic density of odd order reductions |
$9/56$ |