| Curve name |
$X_{297}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$1$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 10 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 9 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 3 \\ 2 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{89}$ |
| Curves that $X_{297}$ minimally covers |
$X_{89}$, $X_{103}$, $X_{167}$ |
| Curves that minimally cover $X_{297}$ |
$X_{555}$, $X_{556}$, $X_{557}$, $X_{558}$ |
| Curves that minimally cover $X_{297}$ and have infinitely many rational
points. |
|
| Model |
\[y^2 = x^3 + x^2 - 9x + 7\] |
| Info about rational points |
$X_{297}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$ |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 6740460588x - 200219177111472$, with conductor
$1993828200$ |
| Generic density of odd order reductions |
$2193/7168$ |