| Curve name |
$X_{165}$ |
| Index |
$24$ |
| Level |
$16$ |
| Genus |
$1$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 13 & 10 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 13 & 13 \\ 2 & 3 \end{matrix}\right],
\left[ \begin{matrix} 15 & 13 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 11 & 11 \\ 2 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{39}$ |
| Curves that $X_{165}$ minimally covers |
$X_{39}$ |
| Curves that minimally cover $X_{165}$ |
$X_{288}$, $X_{308}$, $X_{358}$, $X_{359}$, $X_{381}$, $X_{390}$, $X_{395}$, $X_{399}$, $X_{400}$, $X_{402}$ |
| Curves that minimally cover $X_{165}$ and have infinitely many rational
points. |
$X_{288}$, $X_{308}$ |
| Model |
\[y^2 = x^3 + x^2 - 13x - 21\] |
| Info about rational points |
$X_{165}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$ |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
None. All the rational points lift to covering modular curves. |
| Generic density of odd order reductions |
N/A |