| Curve name |
$X_{563}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$3$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 14 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 5 \\ 6 & 11 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{295}$ |
| Curves that $X_{563}$ minimally covers |
$X_{295}$ |
| Curves that minimally cover $X_{563}$ |
|
| Curves that minimally cover $X_{563}$ and have infinitely many rational
points. |
|
| Model |
\[y^2 = x^7 + 4x^6 - 7x^5 - 8x^4 + 7x^3 + 4x^2 - x\] |
| Info about rational points |
| Rational point | Image on the $j$-line |
| $(1 : 0 : 0)$ |
\[2048\]
|
| $(-1 : 0 : 1)$ |
\[2048\]
|
| $(0 : 0 : 1)$ |
\[2048\]
|
| $(1 : 0 : 1)$ |
\[2048\]
|
|
| Comments on finding rational points |
This curve is isomorphic to $X_{556}$. |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 3x - 2$, with conductor $200$ |
| Generic density of odd order reductions |
$2195/7168$ |