| Curve name |
$X_{619}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$3$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 25 & 18 \\ 2 & 7 \end{matrix}\right],
\left[ \begin{matrix} 25 & 25 \\ 2 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 25 & 11 \\ 2 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{288}$ |
| Curves that $X_{619}$ minimally covers |
$X_{288}$ |
| Curves that minimally cover $X_{619}$ |
|
| Curves that minimally cover $X_{619}$ and have infinitely many rational
points. |
|
| Model |
\[x^4 - x^2y^2 - 2x^2z^2 - y^3z + 2yz^3 = 0\] |
| Info about rational points |
| Rational point | Image on the $j$-line |
| $(-2 : -4 : 1)$ |
\[\frac{-631595585199146625}{218340105584896}\]
|
| $(0 : 1 : 0)$ |
\[ \infty \]
|
| $(0 : 0 : 1)$ |
\[ \infty \]
|
| $(1 : 1 : 0)$ |
\[16581375 \,\,(\text{CM by }-28)\]
|
| $(2 : -4 : 1)$ |
\[\frac{-631595585199146625}{218340105584896}\]
|
| $(-1 : 1 : 0)$ |
\[16581375 \,\,(\text{CM by }-28)\]
|
|
| Comments on finding rational points |
This curve admits a family of etale double covers of genus 5. These double
covers have 16 automorphisms defined over
$K = \mathbb{Q}(\sqrt{2+\sqrt{2}})$ and as a consequence map to elliptic curves
defined
over $K$. These curves have rank two over $K$ and we use elliptic curve
Chabauty. |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - x^2 - 303870x - 81409651$, with conductor $17918$ |
| Generic density of odd order reductions |
$410657/1835008$ |