| Curve name |
$X_{50}$ |
| 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 & 3 \\ 4 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 6 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{11}$ |
| Curves that $X_{50}$ minimally covers |
$X_{11}$ |
| Curves that minimally cover $X_{50}$ |
$X_{65}$, $X_{69}$, $X_{71}$, $X_{73}$, $X_{90}$, $X_{91}$, $X_{113}$, $X_{114}$, $X_{125}$, $X_{131}$, $X_{134}$, $X_{141}$, $X_{152}$, $X_{154}$, $X_{155}$, $X_{156}$, $X_{170}$, $X_{171}$ |
| Curves that minimally cover $X_{50}$ and have infinitely many rational
points. |
$X_{65}$, $X_{69}$, $X_{71}$, $X_{73}$, $X_{90}$, $X_{91}$, $X_{113}$, $X_{114}$, $X_{155}$, $X_{156}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{50}) = \mathbb{Q}(f_{50}), f_{11} =
f_{50}^{2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 248453x - 47558454$, with conductor $71148$ |
| Generic density of odd order reductions |
$2659/10752$ |