| Curve name |
$X_{39}$ |
| Index |
$12$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 2 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 2 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{11}$ |
| Curves that $X_{39}$ minimally covers |
$X_{11}$ |
| Curves that minimally cover $X_{39}$ |
$X_{63}$, $X_{66}$, $X_{80}$, $X_{83}$, $X_{90}$, $X_{91}$, $X_{95}$, $X_{97}$, $X_{105}$, $X_{106}$, $X_{107}$, $X_{124}$, $X_{128}$, $X_{143}$, $X_{144}$, $X_{147}$, $X_{160}$, $X_{161}$, $X_{165}$, $X_{166}$ |
| Curves that minimally cover $X_{39}$ and have infinitely many rational
points. |
$X_{63}$, $X_{66}$, $X_{80}$, $X_{83}$, $X_{90}$, $X_{91}$, $X_{95}$, $X_{97}$, $X_{105}$, $X_{106}$, $X_{107}$, $X_{124}$, $X_{165}$, $X_{166}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{39}) = \mathbb{Q}(f_{39}), f_{11} =
\frac{-64}{f_{39}^{2} - 8}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 1617x + 18666$, with conductor $1575$ |
| Generic density of odd order reductions |
$2659/10752$ |