| Curve name |
$X_{194}$ |
| 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} 7 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 0 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{96}$ |
| Curves that $X_{194}$ minimally covers |
$X_{96}$, $X_{98}$, $X_{99}$ |
| Curves that minimally cover $X_{194}$ |
$X_{451}$, $X_{460}$, $X_{464}$, $X_{465}$, $X_{468}$, $X_{469}$, $X_{194a}$, $X_{194b}$, $X_{194c}$, $X_{194d}$, $X_{194e}$, $X_{194f}$, $X_{194g}$, $X_{194h}$, $X_{194i}$, $X_{194j}$, $X_{194k}$, $X_{194l}$ |
| Curves that minimally cover $X_{194}$ and have infinitely many rational
points. |
$X_{194a}$, $X_{194b}$, $X_{194c}$, $X_{194d}$, $X_{194e}$, $X_{194f}$, $X_{194g}$, $X_{194h}$, $X_{194i}$, $X_{194j}$, $X_{194k}$, $X_{194l}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{194}) = \mathbb{Q}(f_{194}), f_{96} =
\frac{f_{194}^{2} + \frac{1}{4}}{f_{194}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 64980x + 5986876$, with conductor $1530$ |
| Generic density of odd order reductions |
$25/224$ |