| Curve name |
$X_{33}$ |
| 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} 3 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 4 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{13}$ |
| Curves that $X_{33}$ minimally covers |
$X_{13}$ |
| Curves that minimally cover $X_{33}$ |
$X_{78}$, $X_{100}$, $X_{33a}$, $X_{33b}$, $X_{33c}$, $X_{33d}$, $X_{33e}$, $X_{33f}$, $X_{33g}$, $X_{33h}$ |
| Curves that minimally cover $X_{33}$ and have infinitely many rational
points. |
$X_{78}$, $X_{100}$, $X_{33a}$, $X_{33b}$, $X_{33c}$, $X_{33d}$, $X_{33e}$, $X_{33f}$, $X_{33g}$, $X_{33h}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{33}) = \mathbb{Q}(f_{33}), f_{13} =
-2f_{33}^{2} - 8\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 18x + 4$, with conductor $198$ |
| Generic density of odd order reductions |
$513/3584$ |