| Curve name |
$X_{395}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$2$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 6 \\ 2 & 9 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 10 & 13 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 2 & 9 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{66}$ |
| Curves that $X_{395}$ minimally covers |
$X_{66}$, $X_{165}$, $X_{166}$ |
| Curves that minimally cover $X_{395}$ |
|
| Curves that minimally cover $X_{395}$ and have infinitely many rational
points. |
|
| Model |
\[y^2 = x^6 - 5x^4 - 5x^2 + 1\] |
| Info about rational points |
| Rational point | Image on the $j$-line |
| $(1 : -1 : 0)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
| $(1 : 1 : 0)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
| $(0 : -1 : 1)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
| $(0 : 1 : 1)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
|
| Comments on finding rational points |
The rank of the Jacobian is 2. This curve admits a family of etale double
covers that map to rank zero elliptic curves. |
| Elliptic curve whose $2$-adic image is the subgroup |
None |
| Generic density of odd order reductions |
N/A |