| Curve name |
$X_{520}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$2$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 29 & 26 \\ 2 & 3 \end{matrix}\right],
\left[ \begin{matrix} 29 & 29 \\ 2 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{216}$ |
| Curves that $X_{520}$ minimally covers |
$X_{216}$ |
| Curves that minimally cover $X_{520}$ |
|
| Curves that minimally cover $X_{520}$ and have infinitely many rational
points. |
|
| Model |
\[y^2 = x^6 + 8x^5 + 6x^4 + 12x^2 - 32x + 8\] |
| Info about rational points |
| Rational point | Image on the $j$-line |
| $(1 : -1 : 0)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
| $(1 : 1 : 0)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
|
| Comments on finding rational points |
The rank of the Jacobian is 2. We construct a family of etale double covers,
but one of these maps to an elliptic curve of rank 1. We construct an etale
four-fold cover over $\mathbb{Q}(\sqrt{2})$ that maps to an elliptic curve over
$\mathbb{Q}(\sqrt{2})$ and use elliptic curve Chabauty. |
| Elliptic curve whose $2$-adic image is the subgroup |
None |
| Generic density of odd order reductions |
N/A |