| Curve name |
$X_{19}$ |
| Index |
$6$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 2 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 0 & 5 \end{matrix}\right]$ |
| Images in lower levels |
| Level | Index of image | Corresponding curve |
| $2$ |
$3$ |
$X_{6}$ |
| $4$ |
$3$ |
$X_{6}$ |
|
| Meaning/Special name |
Elliptic curves that acquire full $2$-torsion over
$\mathbb{Q}(\sqrt{-2})$ |
| Chosen covering |
$X_{6}$ |
| Curves that $X_{19}$ minimally covers |
$X_{4}$, $X_{6}$ |
| Curves that minimally cover $X_{19}$ |
$X_{29}$, $X_{48}$ |
| Curves that minimally cover $X_{19}$ and have infinitely many rational
points. |
$X_{29}$, $X_{48}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{19}) = \mathbb{Q}(f_{19}), f_{6} =
\frac{48f_{19}^{2} - 32}{f_{19}^{2} + 2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 + 4x + 20$, with conductor $66$ |
| Generic density of odd order reductions |
$5123/21504$ |