| Curve name |
$X_{618}$ |
| 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} 25 & 25 \\ 2 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 29 & 26 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{288}$ |
| Curves that $X_{618}$ minimally covers |
$X_{288}$ |
| Curves that minimally cover $X_{618}$ |
|
| Curves that minimally cover $X_{618}$ and have infinitely many rational
points. |
|
| Model |
\[x^4 + x^2y^2 + 2x^2z^2 - y^3z + 2yz^3 = 0\] |
| Info about rational points |
| Rational point | Image on the $j$-line |
| $(0 : 1 : 0)$ |
\[ \infty \]
|
| $(0 : 0 : 1)$ |
\[ \infty \]
|
|
| Comments on finding rational points |
This curve maps to a rank zero elliptic curve defined over
$\mathbb{Q}(\sqrt{2})$. |
| Elliptic curve whose $2$-adic image is the subgroup |
None |
| Generic density of odd order reductions |
N/A |