| Curve name |
$X_{675}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$5$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 25 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 13 & 7 \\ 2 & 3 \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 $X_{675}$ minimally covers |
$X_{291}$ |
| Curves that minimally cover $X_{675}$ |
|
| Curves that minimally cover $X_{675}$ and have infinitely many rational
points. |
|
| Model |
\[y^2 = x^3 - 2x\]\[w^2 = -4x^2y - 4xy - y^3\] |
| Info about rational points |
| Rational point | Image on the $j$-line |
| $(-1 : 1 : 1 : 1)$ |
\[16581375 \,\,(\text{CM by }-28)\]
|
| $(0 : 0 : 1 : 0)$ |
Singular
|
| $(0 : 0 : 0 : 1)$ |
Singular
|
| $(1 : -1 : -1 : 1)$ |
\[16581375 \,\,(\text{CM by }-28)\]
|
|
| Comments on finding rational points |
This curve admits a family of twists of etale double covers that are also
modular curves. Each of these modular curves maps to one we have already
computed that has finitely many rational points. |
| Elliptic curve whose $2$-adic image is the subgroup |
None |
| Generic density of odd order reductions |
N/A |