| Curve name |
$X_{634}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$3$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 2 \\ 20 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 6 \\ 20 & 29 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 10 & 31 \end{matrix}\right],
\left[ \begin{matrix} 1 & 5 \\ 28 & 15 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{308}$ |
| Curves that $X_{634}$ minimally covers |
$X_{308}$ |
| Curves that minimally cover $X_{634}$ |
|
| Curves that minimally cover $X_{634}$ 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 |
| $(-2 : -4 : 1)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
| $(0 : 1 : 0)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
| $(0 : 0 : 1)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
| $(1 : 1 : 0)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
| $(2 : -4 : 1)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
| $(-1 : 1 : 0)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
|
| Comments on finding rational points |
This curve is isomorphic to $X_{619}$. |
| Elliptic curve whose $2$-adic image is the subgroup |
None |
| Generic density of odd order reductions |
N/A |