| Curve name |
$X_{688}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$5$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 2 \\ 18 & 11 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 10 & 15 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 20 & 21 \end{matrix}\right],
\left[ \begin{matrix} 1 & 7 \\ 26 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{349}$ |
| Curves that $X_{688}$ minimally covers |
$X_{349}$ |
| Curves that minimally cover $X_{688}$ |
|
| Curves that minimally cover $X_{688}$ and have infinitely many rational
points. |
|
| Model |
\[y^2 = x^3 + x^2 - 13x - 21\]\[w^2 = 98x^2y^2 + 3552x^2y - 12032x^2 -
1140xy^2 + 8448xy - 43008x - 140y^3 - 5054y^2 + 2080y - 33024\] |
| Info about rational points |
| Rational point | Image on the $j$-line |
| $(-1/288 : 1/180 : -1/1440 : 1)$ |
\[0 \,\,(\text{CM by }-3)\]
|
| $(-115/49 : 496/343 : 1 : 0)$ |
Singular
|
| $(1/288 : -1/180 : 1/1440 : 1)$ |
\[0 \,\,(\text{CM by }-3)\]
|
| $(0 : 0 : 0 : 1)$ |
Singular
|
|
| 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 |