| Curve name |
$X_{41}$ |
| Index |
$12$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 2 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 2 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{11}$ |
| Curves that $X_{41}$ minimally covers |
$X_{11}$, $X_{15}$, $X_{17}$ |
| Curves that minimally cover $X_{41}$ |
$X_{80}$, $X_{81}$, $X_{82}$, $X_{131}$, $X_{138}$ |
| Curves that minimally cover $X_{41}$ and have infinitely many rational
points. |
$X_{80}$, $X_{81}$, $X_{82}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{41}) = \mathbb{Q}(f_{41}), f_{11} =
\frac{8f_{41}^{2} + 16}{f_{41}^{2} - 2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 1591x - 18317$, with conductor $17664$ |
| Generic density of odd order reductions |
$2659/10752$ |