| Curve name |
$X_{324}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$1$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 2 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 10 & 15 \end{matrix}\right],
\left[ \begin{matrix} 1 & 5 \\ 10 & 15 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{69}$ |
| Curves that $X_{324}$ minimally covers |
$X_{69}$, $X_{113}$, $X_{156}$ |
| Curves that minimally cover $X_{324}$ |
$X_{568}$, $X_{569}$, $X_{570}$, $X_{575}$, $X_{654}$, $X_{693}$, $X_{694}$, $X_{712}$ |
| Curves that minimally cover $X_{324}$ and have infinitely many rational
points. |
|
| Model |
\[y^2 = x^3 - 2x\] |
| Info about rational points |
$X_{324}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$ |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 + x^2 - 3503663358x + 159400872404283$, with conductor
$2665869738$ |
| Generic density of odd order reductions |
$12833/57344$ |