| Curve name |
$X_{45}$ |
| Index |
$12$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 5 \\ 2 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 5 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{11}$ |
| Curves that $X_{45}$ minimally covers |
$X_{11}$ |
| Curves that minimally cover $X_{45}$ |
$X_{61}$, $X_{69}$, $X_{70}$, $X_{73}$, $X_{77}$, $X_{81}$, $X_{94}$, $X_{97}$, $X_{109}$, $X_{110}$, $X_{111}$, $X_{112}$, $X_{130}$, $X_{136}$, $X_{140}$, $X_{143}$, $X_{149}$, $X_{150}$, $X_{151}$, $X_{153}$ |
| Curves that minimally cover $X_{45}$ and have infinitely many rational
points. |
$X_{61}$, $X_{69}$, $X_{70}$, $X_{73}$, $X_{77}$, $X_{81}$, $X_{94}$, $X_{97}$, $X_{109}$, $X_{110}$, $X_{111}$, $X_{112}$, $X_{150}$, $X_{153}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{45}) = \mathbb{Q}(f_{45}), f_{11} =
-f_{45}^{2} + 8\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 + 258x + 1791$, with conductor $1575$ |
| Generic density of odd order reductions |
$2659/10752$ |