| Curve name |
$X_{188}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 6 \\ 0 & 1 \end{matrix}\right],
\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]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{98}$ |
| Curves that $X_{188}$ minimally covers |
$X_{98}$, $X_{100}$, $X_{101}$ |
| Curves that minimally cover $X_{188}$ |
$X_{449}$, $X_{450}$, $X_{455}$, $X_{462}$, $X_{188a}$, $X_{188b}$, $X_{188c}$, $X_{188d}$, $X_{188e}$, $X_{188f}$, $X_{188g}$, $X_{188h}$ |
| Curves that minimally cover $X_{188}$ and have infinitely many rational
points. |
$X_{188a}$, $X_{188b}$, $X_{188c}$, $X_{188d}$, $X_{188e}$, $X_{188f}$, $X_{188g}$, $X_{188h}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{188}) = \mathbb{Q}(f_{188}), f_{98} =
\frac{f_{188}^{2} + 2}{f_{188}^{2} - 2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 756x - 7808$, with conductor $126$ |
| Generic density of odd order reductions |
$193/1792$ |