| Curve name |
$X_{218}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 14 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 10 & 15 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{81}$ |
| Curves that $X_{218}$ minimally covers |
$X_{81}$, $X_{110}$, $X_{111}$ |
| Curves that minimally cover $X_{218}$ |
|
| Curves that minimally cover $X_{218}$ and have infinitely many rational
points. |
|
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{218}) = \mathbb{Q}(f_{218}), f_{81} =
\frac{f_{218}}{f_{218}^{2} + \frac{1}{8}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 3551240x + 2575835648$, with conductor $147712$ |
| Generic density of odd order reductions |
$45667/172032$ |