| Curve name |
$X_{214}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 0 \\ 0 & 13 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 2 & 9 \end{matrix}\right],
\left[ \begin{matrix} 1 & 4 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{66}$ |
| Curves that $X_{214}$ minimally covers |
$X_{66}$, $X_{105}$, $X_{107}$ |
| Curves that minimally cover $X_{214}$ |
$X_{214a}$, $X_{214b}$, $X_{214c}$, $X_{214d}$ |
| Curves that minimally cover $X_{214}$ and have infinitely many rational
points. |
$X_{214a}$, $X_{214b}$, $X_{214c}$, $X_{214d}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{214}) = \mathbb{Q}(f_{214}), f_{66} =
\frac{8f_{214} + 8}{f_{214}^{2} - 2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 10706425x - 10586268000$, with conductor $421600$ |
| Generic density of odd order reductions |
$9249/57344$ |