| Curve name |
$X_{7}$ |
| Index |
$4$ |
| Level |
$4$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 2 & 1 \\ 3 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 1 \\ 2 & 1 \end{matrix}\right]$ |
| Images in lower levels |
| Level | Index of image | Corresponding curve |
| $2$ |
$1$ |
$X_{1}$ |
|
| Meaning/Special name |
Elliptic curves with $a_{p}(E) \equiv 0 \pmod{4}$ if $p$ is inert in
$\mathbb{Q}(\sqrt{\Delta})$ |
| Chosen covering |
$X_{1}$ |
| Curves that $X_{7}$ minimally covers |
$X_{1}$ |
| Curves that minimally cover $X_{7}$ |
$X_{20}$, $X_{21}$, $X_{22}$, $X_{26}$, $X_{55}$ |
| Curves that minimally cover $X_{7}$ and have infinitely many rational
points. |
$X_{20}$, $X_{22}$, $X_{26}$, $X_{55}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{7}) = \mathbb{Q}(f_{7}), f_{1} =
\frac{32f_{7} - 4}{f_{7}^{4}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 12x + 20$, with conductor $216$ |
| Generic density of odd order reductions |
$179/336$ |