| Curve name |
$X_{192}$ |
| 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 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 6 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{85}$ |
| Curves that $X_{192}$ minimally covers |
$X_{85}$, $X_{96}$, $X_{101}$ |
| Curves that minimally cover $X_{192}$ |
$X_{451}$, $X_{458}$, $X_{482}$, $X_{483}$, $X_{504}$, $X_{505}$, $X_{506}$, $X_{519}$, $X_{192a}$, $X_{192b}$, $X_{192c}$, $X_{192d}$, $X_{192e}$, $X_{192f}$, $X_{192g}$, $X_{192h}$, $X_{192i}$, $X_{192j}$, $X_{192k}$, $X_{192l}$, $X_{192m}$, $X_{192n}$ |
| Curves that minimally cover $X_{192}$ and have infinitely many rational
points. |
$X_{192a}$, $X_{192b}$, $X_{192c}$, $X_{192d}$, $X_{192e}$, $X_{192f}$, $X_{192g}$, $X_{192h}$, $X_{192i}$, $X_{192j}$, $X_{192k}$, $X_{192l}$, $X_{192m}$, $X_{192n}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{192}) = \mathbb{Q}(f_{192}), f_{85} =
\frac{8f_{192}}{f_{192}^{2} + 4}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - 14526053x + 21308093948$, with conductor $25410$ |
| Generic density of odd order reductions |
$17/168$ |