| Curve name |
$X_{17}$ |
| 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 & 7 \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 $X_{17}$ minimally covers |
$X_{5}$, $X_{6}$ |
| Curves that minimally cover $X_{17}$ |
$X_{37}$, $X_{40}$, $X_{41}$, $X_{44}$ |
| Curves that minimally cover $X_{17}$ and have infinitely many rational
points. |
$X_{37}$, $X_{40}$, $X_{41}$, $X_{44}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{17}) = \mathbb{Q}(f_{17}), f_{6} =
\frac{48f_{17}^{2} + 32}{f_{17}^{2} - 2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - 36x - 70$, with conductor $14$ |
| Generic density of odd order reductions |
$5123/21504$ |