| Curve name |
$X_{9}$ |
| Index |
$6$ |
| Level |
$4$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 2 & 1 \end{matrix}\right]$ |
| Images in lower levels |
| Level | Index of image | Corresponding curve |
| $2$ |
$3$ |
$X_{6}$ |
|
| Meaning/Special name |
Elliptic curves that acquire a cyclic $4$-isogeny over $\mathbb{Q}(i)$ |
| Chosen covering |
$X_{6}$ |
| Curves that $X_{9}$ minimally covers |
$X_{6}$ |
| Curves that minimally cover $X_{9}$ |
$X_{24}$, $X_{37}$, $X_{9a}$, $X_{9b}$, $X_{9c}$, $X_{9d}$ |
| Curves that minimally cover $X_{9}$ and have infinitely many rational
points. |
$X_{24}$, $X_{37}$, $X_{9a}$, $X_{9b}$, $X_{9c}$, $X_{9d}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{9}) = \mathbb{Q}(f_{9}), f_{6} = f_{9}^{2} +
48\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 12x - 11$, with conductor $180$ |
| Generic density of odd order reductions |
$83/336$ |