| Curve name |
$X_{217}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{75}$ |
| Curves that $X_{217}$ minimally covers |
$X_{75}$, $X_{120}$, $X_{122}$ |
| Curves that minimally cover $X_{217}$ |
$X_{466}$, $X_{467}$, $X_{477}$, $X_{481}$, $X_{217a}$, $X_{217b}$, $X_{217c}$, $X_{217d}$, $X_{217e}$, $X_{217f}$, $X_{217g}$, $X_{217h}$ |
| Curves that minimally cover $X_{217}$ and have infinitely many rational
points. |
$X_{217a}$, $X_{217b}$, $X_{217c}$, $X_{217d}$, $X_{217e}$, $X_{217f}$, $X_{217g}$, $X_{217h}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{217}) = \mathbb{Q}(f_{217}), f_{75} =
\frac{f_{217}}{f_{217}^{2} - 2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 + 3474x - 31010$, with conductor $126$ |
| Generic density of odd order reductions |
$193/1792$ |