| Curve name |
$X_{44}$ |
| 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} 1 & 1 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{13}$ |
| Curves that $X_{44}$ minimally covers |
$X_{13}$, $X_{14}$, $X_{17}$ |
| Curves that minimally cover $X_{44}$ |
$X_{78}$, $X_{79}$, $X_{44a}$, $X_{44b}$, $X_{44c}$, $X_{44d}$ |
| Curves that minimally cover $X_{44}$ and have infinitely many rational
points. |
$X_{78}$, $X_{79}$, $X_{44a}$, $X_{44b}$, $X_{44c}$, $X_{44d}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{44}) = \mathbb{Q}(f_{44}), f_{13} =
\frac{8f_{44}^{2} + 16}{f_{44}^{2} - 2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 6692x - 209034$, with conductor $350$ |
| Generic density of odd order reductions |
$513/3584$ |