| Curve name |
$X_{526}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$2$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 19 & 19 \\ 2 & 1 \end{matrix}\right],
\left[ \begin{matrix} 15 & 0 \\ 2 & 1 \end{matrix}\right],
\left[ \begin{matrix} 29 & 0 \\ 2 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{224}$ |
| Curves that $X_{526}$ minimally covers |
$X_{224}$ |
| Curves that minimally cover $X_{526}$ |
|
| Curves that minimally cover $X_{526}$ and have infinitely many rational
points. |
|
| Model |
\[y^2 = x^6 + 8x^5 + 6x^4 + 12x^2 - 32x + 8\] |
| Info about rational points |
| Rational point | Image on the $j$-line |
| $(1 : -1 : 0)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
| $(1 : 1 : 0)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
|
| Comments on finding rational points |
This curve is isomorphic to $X_{520}$. |
| Elliptic curve whose $2$-adic image is the subgroup |
None |
| Generic density of odd order reductions |
N/A |