| Curve name |
$X_{191}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 7 \\ 2 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{64}$ |
| Curves that $X_{191}$ minimally covers |
$X_{64}$ |
| Curves that minimally cover $X_{191}$ |
$X_{443}$, $X_{445}$, $X_{447}$, $X_{461}$ |
| Curves that minimally cover $X_{191}$ and have infinitely many rational
points. |
|
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{191}) = \mathbb{Q}(f_{191}), f_{64} =
\frac{\frac{1}{2}f_{191}^{2} + 4}{f_{191}^{2} + 8f_{191} + 8}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 + x^2 + 122x - 10940$, with conductor $294$ |
| Generic density of odd order reductions |
$269/1344$ |