| Curve name |
$X_{49}$ |
| Index |
$12$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 2 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 2 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 2 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{11}$ |
| Curves that $X_{49}$ minimally covers |
$X_{11}$ |
| Curves that minimally cover $X_{49}$ |
$X_{70}$, $X_{71}$, $X_{76}$, $X_{95}$, $X_{130}$, $X_{137}$ |
| Curves that minimally cover $X_{49}$ and have infinitely many rational
points. |
$X_{70}$, $X_{71}$, $X_{76}$, $X_{95}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{49}) = \mathbb{Q}(f_{49}), f_{11} =
\frac{64}{f_{49}^{2} + 8}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 + 292x + 9588$, with conductor $300$ |
| Generic density of odd order reductions |
$2659/10752$ |