| Curve name |
$X_{680}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$5$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 9 & 18 \\ 2 & 7 \end{matrix}\right],
\left[ \begin{matrix} 15 & 13 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 17 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 29 & 26 \\ 8 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{281}$ |
| Curves that $X_{680}$ minimally covers |
$X_{281}$ |
| Curves that minimally cover $X_{680}$ |
|
| Curves that minimally cover $X_{680}$ and have infinitely many rational
points. |
|
| Model |
\[y^2 = x^3 + x^2 - 13x - 21\]\[w^2 = 147x^2y^2 + 2448x^2y - 28544x^2 -
702xy^2 + 11392xy - 135680x + 182y^3 - 6797y^2 + 12912y - 152192\] |
| Info about rational points |
| Rational point | Image on the $j$-line |
| $(-115/49 : 496/343 : 1 : 0)$ |
Singular
|
| $(0 : 0 : 0 : 1)$ |
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 |