| Curve name |
$X_{4}$ |
| Index |
$2$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 1 & 1 \end{matrix}\right]$ |
| Images in lower levels |
| Level | Index of image | Corresponding curve |
| $2$ |
$1$ |
$X_{1}$ |
| $4$ |
$1$ |
$X_{1}$ |
|
| Meaning/Special name |
Elliptic curves whose discriminant is minus twice a square |
| Chosen covering |
$X_{1}$ |
| Curves that $X_{4}$ minimally covers |
$X_{1}$ |
| Curves that minimally cover $X_{4}$ |
$X_{19}$, $X_{22}$ |
| Curves that minimally cover $X_{4}$ and have infinitely many rational
points. |
$X_{19}$, $X_{22}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{4}) = \mathbb{Q}(f_{4}), f_{1} = -2f_{4}^{2}
+ 1728\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - x - 2$, with conductor $50$ |
| Generic density of odd order reductions |
$3755/7168$ |