| Curve name |
$X_{161}$ |
| Index |
$24$ |
| Level |
$16$ |
| Genus |
$1$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 15 \\ 2 & 3 \end{matrix}\right],
\left[ \begin{matrix} 13 & 13 \\ 2 & 3 \end{matrix}\right],
\left[ \begin{matrix} 11 & 11 \\ 2 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 10 \\ 2 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{39}$ |
| Curves that $X_{161}$ minimally covers |
$X_{39}$ |
| Curves that minimally cover $X_{161}$ |
$X_{287}$, $X_{301}$, $X_{360}$, $X_{365}$, $X_{370}$, $X_{372}$, $X_{379}$, $X_{394}$, $X_{398}$, $X_{400}$ |
| Curves that minimally cover $X_{161}$ and have infinitely many rational
points. |
|
| Model |
\[y^2 = x^3 - x^2 - 3x - 1\] |
| Info about rational points |
| Rational point | Image on the $j$-line |
| $(0 : 1 : 0)$ |
\[ \infty \]
|
| $(-1 : 0 : 1)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
|
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
None |
| Generic density of odd order reductions |
N/A |