| Curve name |
$X_{641}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$3$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 25 & 18 \\ 2 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 13 & 26 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 7 \\ 2 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{281}$ |
| Curves that $X_{641}$ minimally covers |
$X_{281}$ |
| Curves that minimally cover $X_{641}$ |
|
| Curves that minimally cover $X_{641}$ and have infinitely many rational
points. |
|
| Model |
\[x^4 - 4x^2y^2 - 4x^2z^2 + 3y^4 + 2y^3z + 4y^2z^2 + 2yz^3 + z^4 = 0\] |
| Info about rational points |
| Rational point | Image on the $j$-line |
| $(1 : 1 : 0)$ |
\[54000 \,\,(\text{CM by }-12)\]
|
| $(-1 : 1 : 0)$ |
\[54000 \,\,(\text{CM by }-12)\]
|
|
| Comments on finding rational points |
This curve is isomorphic to $X_{633}$. |
| Elliptic curve whose $2$-adic image is the subgroup |
None |
| Generic density of odd order reductions |
N/A |