| Curve name |
$X_{199}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{75}$ |
| Curves that $X_{199}$ minimally covers |
$X_{75}$, $X_{84}$, $X_{85}$ |
| Curves that minimally cover $X_{199}$ |
$X_{466}$, $X_{477}$, $X_{199a}$, $X_{199b}$, $X_{199c}$, $X_{199d}$, $X_{199e}$, $X_{199f}$, $X_{199g}$, $X_{199h}$ |
| Curves that minimally cover $X_{199}$ and have infinitely many rational
points. |
$X_{199a}$, $X_{199b}$, $X_{199c}$, $X_{199d}$, $X_{199e}$, $X_{199f}$, $X_{199g}$, $X_{199h}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{199}) = \mathbb{Q}(f_{199}), f_{75} =
\frac{f_{199}}{f_{199}^{2} + 2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 14796x + 835434$, with conductor $306$ |
| Generic density of odd order reductions |
$635/5376$ |