| Curve name |
$X_{193}$ |
| 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} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
$\tilde{X}_{1}(2,8)$ |
| Chosen covering |
$X_{96}$ |
| Curves that $X_{193}$ minimally covers |
$X_{96}$, $X_{98}$, $X_{102}$ |
| Curves that minimally cover $X_{193}$ |
$X_{456}$, $X_{465}$, $X_{470}$, $X_{475}$, $X_{507}$, $X_{508}$, $X_{509}$, $X_{510}$, $X_{193a}$, $X_{193b}$, $X_{193c}$, $X_{193d}$, $X_{193e}$, $X_{193f}$, $X_{193g}$, $X_{193h}$, $X_{193i}$, $X_{193j}$, $X_{193k}$, $X_{193l}$, $X_{193m}$, $X_{193n}$ |
| Curves that minimally cover $X_{193}$ and have infinitely many rational
points. |
$X_{193a}$, $X_{193b}$, $X_{193c}$, $X_{193d}$, $X_{193e}$, $X_{193f}$, $X_{193g}$, $X_{193h}$, $X_{193i}$, $X_{193j}$, $X_{193k}$, $X_{193l}$, $X_{193m}$, $X_{193n}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{193}) = \mathbb{Q}(f_{193}), f_{96} =
\frac{f_{193}}{f_{193}^{2} - \frac{1}{4}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - 129473x - 10527244$, with conductor $25410$ |
| Generic density of odd order reductions |
$17/168$ |