| Curve name |
$X_{219}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{85}$ |
| Curves that $X_{219}$ minimally covers |
$X_{85}$, $X_{117}$, $X_{119}$ |
| Curves that minimally cover $X_{219}$ |
$X_{473}$, $X_{483}$, $X_{219a}$, $X_{219b}$, $X_{219c}$, $X_{219d}$, $X_{219e}$, $X_{219f}$, $X_{219g}$, $X_{219h}$ |
| Curves that minimally cover $X_{219}$ and have infinitely many rational
points. |
$X_{219a}$, $X_{219b}$, $X_{219c}$, $X_{219d}$, $X_{219e}$, $X_{219f}$, $X_{219g}$, $X_{219h}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{219}) = \mathbb{Q}(f_{219}), f_{85} =
\frac{f_{219}^{2} + 2f_{219} - 1}{f_{219}^{2} + 1}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 1608x + 24288$, with conductor $600$ |
| Generic density of odd order reductions |
$635/5376$ |