| Curve name |
$X_{628}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$3$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 29 & 26 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 9 & 9 \\ 2 & 7 \end{matrix}\right],
\left[ \begin{matrix} 29 & 29 \\ 2 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 10 \\ 2 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{291}$ |
| Curves that $X_{628}$ minimally covers |
$X_{291}$ |
| Curves that minimally cover $X_{628}$ |
|
| Curves that minimally cover $X_{628}$ 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 |
| $(1 : 1 : 1)$ |
\[16581375 \,\,(\text{CM by }-28)\]
|
| $(0 : 1 : 0)$ |
\[ \infty \]
|
| $(1 : 0 : 0)$ |
\[ \infty \]
|
| $(-1 : -1 : 1)$ |
\[16581375 \,\,(\text{CM by }-28)\]
|
|
| Comments on finding rational points |
We use the simple form of the equation to reduce the problem of finding
rational points on this curve to solving $x^{4} + y^{4} = 2z^{4}$ and $x^{4} +
2y^{4} = z^{4}$. These classical problems have already been treated. |
| Elliptic curve whose $2$-adic image is the subgroup |
None |
| Generic density of odd order reductions |
N/A |