| Curve name |
$X_{691}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$5$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 5 \\ 0 & 31 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 20 & 21 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 30 & 31 \end{matrix}\right],
\left[ \begin{matrix} 1 & 4 \\ 20 & 21 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{323}$ |
| Curves that $X_{691}$ minimally covers |
$X_{323}$ |
| Curves that minimally cover $X_{691}$ |
|
| Curves that minimally cover $X_{691}$ and have infinitely many rational
points. |
|
| Model |
\[y^2 = x^3 + x^2 - 3x + 1\]\[w^2 = -6x^2y - 16x^2 + 8xy + 16x + y^3 + 4y^2
+ 2y\] |
| Info about rational points |
| Rational point | Image on the $j$-line |
| $(1 : 0 : 1 : 0)$ |
\[ \infty \]
|
| $(-1 : -2 : 1 : 0)$ |
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 |