| Curve name |
$X_{189}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 6 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 4 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 4 \\ 4 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{58}$ |
| Curves that $X_{189}$ minimally covers |
$X_{58}$, $X_{98}$, $X_{101}$ |
| Curves that minimally cover $X_{189}$ |
$X_{449}$, $X_{450}$, $X_{458}$, $X_{465}$, $X_{500}$, $X_{501}$, $X_{502}$, $X_{503}$, $X_{189a}$, $X_{189b}$, $X_{189c}$, $X_{189d}$, $X_{189e}$, $X_{189f}$, $X_{189g}$, $X_{189h}$, $X_{189i}$, $X_{189j}$, $X_{189k}$, $X_{189l}$, $X_{189m}$, $X_{189n}$ |
| Curves that minimally cover $X_{189}$ and have infinitely many rational
points. |
$X_{189a}$, $X_{189b}$, $X_{189c}$, $X_{189d}$, $X_{189e}$, $X_{189f}$, $X_{189g}$, $X_{189h}$, $X_{189i}$, $X_{189j}$, $X_{189k}$, $X_{189l}$, $X_{189m}$, $X_{189n}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{189}) = \mathbb{Q}(f_{189}), f_{58} =
\frac{f_{189}^{2} - 8}{f_{189}^{2} + 8f_{189} + 8}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - 913553x + 328508948$, with conductor $25410$ |
| Generic density of odd order reductions |
$17/168$ |