| Curve name |
$X_{626}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$3$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 23 & 19 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 25 & 4 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 15 & 9 \\ 2 & 1 \end{matrix}\right],
\left[ \begin{matrix} 21 & 0 \\ 4 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{350}$ |
| Curves that $X_{626}$ minimally covers |
$X_{350}$ |
| Curves that minimally cover $X_{626}$ |
|
| Curves that minimally cover $X_{626}$ and have infinitely many rational
points. |
|
| Model |
\[-x^4 - y^4 - 2y^3z - 2yz^3 + z^4 = 0\] |
| Info about rational points |
| Rational point | Image on the $j$-line |
| $(-1 : 0 : 1)$ |
\[0 \,\,(\text{CM by }-3)\]
|
| $(1 : 0 : 1)$ |
\[0 \,\,(\text{CM by }-3)\]
|
|
| Comments on finding rational points |
This curve has a family of unramified double
covers in which only one curve has a rational point. This genus 5 curve maps to
a hyperelliptic curve whose Jacobian has rank 1. We use the method of Chabauty
on this curve. |
| Elliptic curve whose $2$-adic image is the subgroup |
None |
| Generic density of odd order reductions |
N/A |