| Curve name |
$X_{407}$ |
| Index |
$48$ |
| Level |
$32$ |
| Genus |
$2$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 9 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 3 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{116}$ |
| Curves that $X_{407}$ minimally covers |
$X_{116}$ |
| Curves that minimally cover $X_{407}$ |
|
| Curves that minimally cover $X_{407}$ and have infinitely many rational
points. |
|
| Model |
\[y^2 = 2x^5 + 2x\] |
| Info about rational points |
| Rational point | Image on the $j$-line |
| $(1 : 0 : 0)$ |
\[ \infty \]
|
| $(0 : 0 : 1)$ |
\[ \infty \]
|
| $(1 : -2 : 1)$ |
\[287496 \,\,(\text{CM by }-16)\]
|
| $(1 : 2 : 1)$ |
\[287496 \,\,(\text{CM by }-16)\]
|
|
| Comments on finding rational points |
The rank of the Jacobian is 1. We use the method of Chabauty. |
| Elliptic curve whose $2$-adic image is the subgroup |
None |
| Generic density of odd order reductions |
N/A |