| Curve name |
$X_{40}$ |
| Index |
$12$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 6 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{12}$ |
| Curves that $X_{40}$ minimally covers |
$X_{12}$, $X_{16}$, $X_{17}$ |
| Curves that minimally cover $X_{40}$ |
$X_{126}$, $X_{145}$, $X_{40a}$, $X_{40b}$, $X_{40c}$, $X_{40d}$ |
| Curves that minimally cover $X_{40}$ and have infinitely many rational
points. |
$X_{40a}$, $X_{40b}$, $X_{40c}$, $X_{40d}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{40}) = \mathbb{Q}(f_{40}), f_{12} =
\frac{8f_{40}^{2} - 1}{f_{40}^{2} + f_{40} + \frac{1}{8}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 318x + 1640$, with conductor $16128$ |
| Generic density of odd order reductions |
$3331/10752$ |