| Curve name |
$X_{637}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$3$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 21 & 20 \\ 2 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 4 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 31 & 30 \\ 2 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 3 \\ 2 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{312}$ |
| Curves that $X_{637}$ minimally covers |
$X_{312}$ |
| Curves that minimally cover $X_{637}$ |
|
| Curves that minimally cover $X_{637}$ and have infinitely many rational
points. |
|
| Model |
\[x^3z + x^2y^2 - 2xz^3 - y^4 + 2y^2z^2 = 0\] |
| Info about rational points |
| Rational point | Image on the $j$-line |
| $(-4 : 2 : 1)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
| $(0 : 0 : 1)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
| $(1 : 0 : 0)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
| $(1 : 1 : 0)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
| $(-4 : -2 : 1)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
| $(-1 : 1 : 0)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
|
| 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 |