| Curve name |
$X_{689}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$5$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 21 & 20 \\ 2 & 3 \end{matrix}\right],
\left[ \begin{matrix} 13 & 0 \\ 2 & 3 \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 $X_{689}$ minimally covers |
$X_{312}$ |
| Curves that minimally cover $X_{689}$ |
|
| Curves that minimally cover $X_{689}$ and have infinitely many rational
points. |
|
| Model |
\[y^2 = x^3 + x^2 - 3x + 1\]\[w^2 = 2x^2y^2 + 8x^2y - 4xy^2 - 8xy - 2y^3 -
2y^2\] |
| Info about rational points |
| Rational point | Image on the $j$-line |
| $(1 : 0 : 1 : 0)$ |
Singular
|
| $(-1 : -2 : 1 : 0)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
| $(0 : -1 : 1 : 0)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
| $(0 : -1/2 : -1/2 : 1)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
| $(0 : 1/2 : 1/2 : 1)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
| $(0 : 0 : 0 : 1)$ |
Singular
|
|
| Comments on finding rational points |
We use the factorization of the elliptic function to construct a family of
etale double covers. Each of these maps to a hyperelliptic curve whose Jacobian
has rank at most 1. We use the method of Chabauty. |
|
| Elliptic curve whose $2$-adic image is the subgroup |
None |
| Generic density of odd order reductions |
N/A |