| Curve name |
$X_{211}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 9 & 0 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 13 & 13 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 9 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 8 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{85}$ |
| Curves that $X_{211}$ minimally covers |
$X_{85}$, $X_{118}$, $X_{122}$ |
| Curves that minimally cover $X_{211}$ |
$X_{472}$, $X_{477}$, $X_{478}$, $X_{482}$, $X_{489}$, $X_{490}$, $X_{511}$, $X_{512}$, $X_{513}$, $X_{514}$, $X_{522}$, $X_{528}$, $X_{530}$, $X_{532}$, $X_{211a}$, $X_{211b}$, $X_{211c}$, $X_{211d}$, $X_{211e}$, $X_{211f}$, $X_{211g}$, $X_{211h}$, $X_{211i}$, $X_{211j}$, $X_{211k}$, $X_{211l}$, $X_{211m}$, $X_{211n}$, $X_{211o}$, $X_{211p}$, $X_{211q}$, $X_{211r}$, $X_{211s}$, $X_{211t}$ |
| Curves that minimally cover $X_{211}$ and have infinitely many rational
points. |
$X_{211a}$, $X_{211b}$, $X_{211c}$, $X_{211d}$, $X_{211e}$, $X_{211f}$, $X_{211g}$, $X_{211h}$, $X_{211i}$, $X_{211j}$, $X_{211k}$, $X_{211l}$, $X_{211m}$, $X_{211n}$, $X_{211o}$, $X_{211p}$, $X_{211q}$, $X_{211r}$, $X_{211s}$, $X_{211t}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{211}) = \mathbb{Q}(f_{211}), f_{85} =
\frac{2f_{211}^{2} + 8}{f_{211}^{2} + 4f_{211} - 4}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - 14435303x + 21587495048$, with conductor $25410$ |
| Generic density of odd order reductions |
$17/168$ |