| Curve name |
$X_{650}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$3$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 25 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 9 & 9 \\ 4 & 3 \end{matrix}\right],
\left[ \begin{matrix} 15 & 13 \\ 2 & 1 \end{matrix}\right],
\left[ \begin{matrix} 29 & 23 \\ 4 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{284}$ |
| Curves that $X_{650}$ minimally covers |
$X_{284}$ |
| Curves that minimally cover $X_{650}$ |
|
| Curves that minimally cover $X_{650}$ 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 |
| $(1 : 1 : 0)$ |
\[54000 \,\,(\text{CM by }-12)\]
|
| $(-1 : 1 : 0)$ |
\[54000 \,\,(\text{CM by }-12)\]
|
|
| Comments on finding rational points |
This curve is isomorphic to $X_{626}$. |
| Elliptic curve whose $2$-adic image is the subgroup |
None |
| Generic density of odd order reductions |
N/A |