| Curve name |
$X_{222}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 12 & 3 \end{matrix}\right],
\left[ \begin{matrix} 9 & 11 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{95}$ |
| Curves that $X_{222}$ minimally covers |
$X_{95}$, $X_{107}$, $X_{124}$ |
| Curves that minimally cover $X_{222}$ |
$X_{222a}$, $X_{222b}$, $X_{222c}$, $X_{222d}$ |
| Curves that minimally cover $X_{222}$ and have infinitely many rational
points. |
$X_{222a}$, $X_{222b}$, $X_{222c}$, $X_{222d}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{222}) = \mathbb{Q}(f_{222}), f_{95} =
\frac{-2}{f_{222}^{2}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 11616x + 594512$, with conductor $4626$ |
| Generic density of odd order reductions |
$9249/57344$ |