| Curve name |
$X_{289}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$1$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 9 & 2 \\ 2 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 2 & 7 \end{matrix}\right],
\left[ \begin{matrix} 9 & 9 \\ 2 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{91}$ |
| Curves that $X_{289}$ minimally covers |
$X_{91}$, $X_{124}$, $X_{166}$ |
| Curves that minimally cover $X_{289}$ |
$X_{582}$, $X_{585}$ |
| Curves that minimally cover $X_{289}$ and have infinitely many rational
points. |
|
| Model |
\[y^2 = x^3 + x^2 - 13x - 21\] |
| Info about rational points |
$X_{289}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$ |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - x^2 + 173490x + 156628973$, with conductor $4046$ |
| Generic density of odd order reductions |
$12833/57344$ |