| Curve name |
$X_{38}$ |
| Index |
$12$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 6 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 6 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 6 & 7 \end{matrix}\right]$ |
| Images in lower levels |
| Level | Index of image | Corresponding curve |
| $2$ |
$6$ |
$X_{8}$ |
| $4$ |
$6$ |
$X_{8}$ |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{8}$ |
| Curves that $X_{38}$ minimally covers |
$X_{8}$, $X_{14}$, $X_{16}$ |
| Curves that minimally cover $X_{38}$ |
$X_{62}$, $X_{66}$, $X_{38a}$, $X_{38b}$, $X_{38c}$, $X_{38d}$ |
| Curves that minimally cover $X_{38}$ and have infinitely many rational
points. |
$X_{62}$, $X_{66}$, $X_{38a}$, $X_{38b}$, $X_{38c}$, $X_{38d}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{38}) = \mathbb{Q}(f_{38}), f_{8} =
f_{38}^{2} - 1\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 442x - 2784$, with conductor $350$ |
| Generic density of odd order reductions |
$513/3584$ |