| Curve name |
$X_{32}$ |
| Index |
$12$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 7 \\ 4 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{13}$ |
| Curves that $X_{32}$ minimally covers |
$X_{13}$ |
| Curves that minimally cover $X_{32}$ |
$X_{79}$, $X_{96}$, $X_{115}$, $X_{116}$, $X_{157}$, $X_{158}$, $X_{32a}$, $X_{32b}$, $X_{32c}$, $X_{32d}$, $X_{32e}$, $X_{32f}$, $X_{32g}$, $X_{32h}$, $X_{32i}$, $X_{32j}$, $X_{32k}$, $X_{32l}$ |
| Curves that minimally cover $X_{32}$ and have infinitely many rational
points. |
$X_{79}$, $X_{96}$, $X_{115}$, $X_{116}$, $X_{32a}$, $X_{32b}$, $X_{32c}$, $X_{32d}$, $X_{32e}$, $X_{32f}$, $X_{32g}$, $X_{32h}$, $X_{32i}$, $X_{32j}$, $X_{32k}$, $X_{32l}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{32}) = \mathbb{Q}(f_{32}), f_{13} =
-f_{32}^{2} - 8\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 195x - 1000$, with conductor $1845$ |
| Generic density of odd order reductions |
$9/56$ |