| Curve name |
$X_{654}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$3$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 20 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 10 & 15 \end{matrix}\right],
\left[ \begin{matrix} 1 & 5 \\ 28 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 4 \\ 20 & 13 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{324}$ |
| Curves that $X_{654}$ minimally covers |
$X_{324}$ |
| Curves that minimally cover $X_{654}$ |
|
| Curves that minimally cover $X_{654}$ and have infinitely many rational
points. |
|
| Model |
\[-x^3y + 2xy^3 - z^4 = 0\] |
| Info about rational points |
| Rational point | Image on the $j$-line |
| $(1 : 1 : 1)$ |
\[-3375 \,\,(\text{CM by }-7)\]
|
| $(0 : 1 : 0)$ |
\[ \infty \]
|
| $(1 : 0 : 0)$ |
\[ \infty \]
|
| $(-1 : -1 : 1)$ |
\[-3375 \,\,(\text{CM by }-7)\]
|
|
| Comments on finding rational points |
This curve is isomorphic to $X_{628}$. |
| Elliptic curve whose $2$-adic image is the subgroup |
None |
| Generic density of odd order reductions |
N/A |