| Curve name |
$X_{558}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$3$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 5 \\ 6 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 5 \\ 14 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 7 \\ 2 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{297}$ |
| Curves that $X_{558}$ minimally covers |
$X_{297}$ |
| Curves that minimally cover $X_{558}$ |
|
| Curves that minimally cover $X_{558}$ and have infinitely many rational
points. |
|
| Model |
\[y^2 = x^8 - 4x^7 - 12x^6 + 28x^5 + 38x^4 - 28x^3 - 12x^2 + 4x + 1\] |
| Info about rational points |
| Rational point | Image on the $j$-line |
| $(1 : -1 : 0)$ |
\[ \infty \]
|
| $(1 : 1 : 0)$ |
\[\frac{68769820673}{16}\]
|
| $(-1 : -4 : 1)$ |
\[ \infty \]
|
| $(-1 : 4 : 1)$ |
\[\frac{68769820673}{16}\]
|
| $(0 : -1 : 1)$ |
\[ \infty \]
|
| $(0 : 1 : 1)$ |
\[\frac{68769820673}{16}\]
|
| $(1 : -4 : 1)$ |
\[ \infty \]
|
| $(1 : 4 : 1)$ |
\[\frac{68769820673}{16}\]
|
|
| Comments on finding rational points |
A family of etale double covers maps to genus 2 hyperelliptic curves whose
Jacobians have rank zero or one. |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 + x^2 - 5882694x - 5494222780$, with conductor $3362$ |
| Generic density of odd order reductions |
$6515/21504$ |