| Curve name |
$X_{46}$ |
| Index |
$12$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 6 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 6 & 7 \end{matrix}\right]$ |
| Images in lower levels |
| Level | Index of image | Corresponding curve |
| $2$ |
$6$ |
$X_{8}$ |
| $4$ |
$6$ |
$X_{8}$ |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{8}$ |
| Curves that $X_{46}$ minimally covers |
$X_{8}$, $X_{15}$, $X_{18}$ |
| Curves that minimally cover $X_{46}$ |
$X_{62}$, $X_{46a}$, $X_{46b}$, $X_{46c}$, $X_{46d}$ |
| Curves that minimally cover $X_{46}$ and have infinitely many rational
points. |
$X_{62}$, $X_{46a}$, $X_{46b}$, $X_{46c}$, $X_{46d}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{46}) = \mathbb{Q}(f_{46}), f_{8} =
-f_{46}^{2} - 1\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 198x + 1120$, with conductor $198$ |
| Generic density of odd order reductions |
$513/3584$ |