| Curve name |
$X_{213}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{102}$ |
| Curves that $X_{213}$ minimally covers |
$X_{102}$, $X_{118}$, $X_{119}$ |
| Curves that minimally cover $X_{213}$ |
$X_{470}$, $X_{472}$, $X_{491}$, $X_{492}$, $X_{213a}$, $X_{213b}$, $X_{213c}$, $X_{213d}$, $X_{213e}$, $X_{213f}$, $X_{213g}$, $X_{213h}$, $X_{213i}$, $X_{213j}$, $X_{213k}$, $X_{213l}$ |
| Curves that minimally cover $X_{213}$ and have infinitely many rational
points. |
$X_{213a}$, $X_{213b}$, $X_{213c}$, $X_{213d}$, $X_{213e}$, $X_{213f}$, $X_{213g}$, $X_{213h}$, $X_{213i}$, $X_{213j}$, $X_{213k}$, $X_{213l}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{213}) = \mathbb{Q}(f_{213}), f_{102} =
f_{213}^{2} + 1\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 1021230x + 397475626$, with conductor $1530$ |
| Generic density of odd order reductions |
$25/224$ |