| Curve name |
$X_{190}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 2 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 4 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 2 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{99}$ |
| Curves that $X_{190}$ minimally covers |
$X_{99}$, $X_{100}$, $X_{101}$ |
| Curves that minimally cover $X_{190}$ |
$X_{446}$, $X_{449}$, $X_{462}$, $X_{463}$, $X_{190a}$, $X_{190b}$, $X_{190c}$, $X_{190d}$, $X_{190e}$, $X_{190f}$, $X_{190g}$, $X_{190h}$ |
| Curves that minimally cover $X_{190}$ and have infinitely many rational
points. |
$X_{190a}$, $X_{190b}$, $X_{190c}$, $X_{190d}$, $X_{190e}$, $X_{190f}$, $X_{190g}$, $X_{190h}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{190}) = \mathbb{Q}(f_{190}), f_{99} =
\frac{f_{190}^{2} - 8}{f_{190}^{2} + 8f_{190} + 8}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 108x + 288$, with conductor $600$ |
| Generic density of odd order reductions |
$635/5376$ |