| Curve name |
$X_{291}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$1$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 9 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 13 & 10 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 13 & 13 \\ 2 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 5 \\ 8 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{91}$ |
| Curves that $X_{291}$ minimally covers |
$X_{91}$, $X_{114}$, $X_{156}$ |
| Curves that minimally cover $X_{291}$ |
$X_{585}$, $X_{589}$, $X_{591}$, $X_{615}$, $X_{628}$, $X_{675}$, $X_{677}$, $X_{713}$ |
| Curves that minimally cover $X_{291}$ and have infinitely many rational
points. |
|
| Model |
\[y^2 = x^3 - 2x\] |
| Info about rational points |
$X_{291}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$ |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 + x^2 - 70411166718x + 7186776239309115$, with conductor
$2665869738$ |
| Generic density of odd order reductions |
$12833/57344$ |