| Curve name |
$X_{216}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 2 \\ 10 & 15 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 10 & 15 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{77}$ |
| Curves that $X_{216}$ minimally covers |
$X_{77}$, $X_{110}$, $X_{112}$ |
| Curves that minimally cover $X_{216}$ |
$X_{520}$, $X_{533}$ |
| Curves that minimally cover $X_{216}$ and have infinitely many rational
points. |
|
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{216}) = \mathbb{Q}(f_{216}), f_{77} =
\frac{f_{216}^{2} - \frac{1}{8}}{f_{216}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 14413x + 1182613$, with conductor $13056$ |
| Generic density of odd order reductions |
$12833/57344$ |