| Curve name |
$X_{22}$ |
| Index |
$8$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 2 & 3 \\ 1 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 7 \\ 6 & 7 \end{matrix}\right]$ |
| Images in lower levels |
| Level | Index of image | Corresponding curve |
| $2$ |
$1$ |
$X_{1}$ |
| $4$ |
$4$ |
$X_{7}$ |
|
| Meaning/Special name |
Elliptic curves whose discriminant is minus twice a square with $a_{p}(E)
\equiv 0 \pmod{4}$ for $p \equiv 5 \text{ or } 7 \pmod{8}$ |
| Chosen covering |
$X_{7}$ |
| Curves that $X_{22}$ minimally covers |
$X_{4}$, $X_{7}$ |
| Curves that minimally cover $X_{22}$ |
$X_{56}$, $X_{57}$, $X_{83}$, $X_{179}$ |
| Curves that minimally cover $X_{22}$ and have infinitely many rational
points. |
$X_{56}$, $X_{57}$, $X_{83}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{22}) = \mathbb{Q}(f_{22}), f_{7} =
\frac{\frac{1}{6}f_{22}^{2} - 3}{f_{22}^{2} + 12f_{22} + 30}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 88333x + 10222037$, with conductor $44800$ |
| Generic density of odd order reductions |
$955/1792$ |