| Curve name |
$X_{633}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$3$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 29 & 0 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 31 & 27 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 13 & 7 \\ 2 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 4 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{349}$ |
| Curves that $X_{633}$ minimally covers |
$X_{349}$ |
| Curves that minimally cover $X_{633}$ |
|
| Curves that minimally cover $X_{633}$ and have infinitely many rational
points. |
|
| Model |
\[x^4 - 4x^2y^2 - 4x^2z^2 + y^4 + 2y^3z + 4y^2z^2 + 2yz^3 + 3z^4 = 0\] |
| 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 |